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T  R  A  N  S  A  C  T  I  0 


OF   THE 


AMERICAN  PHILOSOPHICAL  SOCIETY, 

1L   ..ajj  HELD  AT  PHILADELPHIA, 

FOR  PROMOTING  USEFUL  KNOWLEDGE,  , 

VOLUME  XIX.-NEW  SERIES. 


PART  I. 

* 


1 .  — A  New  Method  of  Determining   the   General  Perturbations  of  the  Minor  Planets.     By 
William  McKniyht  Bitter,  M.A. 

ARTICLE  IT. — An   Essay  on  the   Development  of  the    Mouth  Parts  of    Certain  Insects.     By  John  B, 
Smith,  Xc.D. 


PUBLISHED  BY  THE   SOCIETY, 

AND   FOR   SALE   BY 

THE   AMERICAN    PHILOSOPHICAL    SOCIETY,    PHILADELPHIA 
N.  TRUBNER  &  CO.,  57  and  59  LUDGATE  HILL,  LONDON. 


ASTRONOMY  LIBRARY 


TRANSACTIONS 


OP    THE 


AMERICAN  PHILOSOPHICAL  SOCIETY 


AETICLE  I. 


A  NEW  METHOD  OF  DETERMINING  THE  GENERAL  PERTURBATIONS  OF 

THE  MINOR  PLANETS. 


BY  WILLIAM  MCKNIGHT  RITTER,  M.A. 

Read  before  the  American  Philosophical  Society,  February  28,  1896. 


PREFACE. 


In  determining  the  general  perturbations  of  the  minor  planets  the  principal  diffi- 
culty arises  from  the  large  eccentricities  and  inclinations  of  these  bodies.  Methods 
that  are  applicable  to  the  major  planets  fail  when  applied  to  the  minor  planets  on 
account  of  want  of  convergence  of  the  series.  For  a  long  time  astronomers  had  to  be 
content  with  finding  what  are  called  the  special  perturbations  of  these  bodies.  And 
it  was  not  until  the  brilliant  researches  of  HANSEIST  on  this  subject  that  serious  hopes 
were  entertained  of  being  able  to  find  also  the  general  perturbations  of  the  minor 
planets.  HANSEN'S  mode  of  treatment  differs  entirely  from  those  that  had  been  pre- 
viously employed.  Instead  of  determining  the  perturbations  of  the  rectangular  or 
polar  coordinates,  or  determining  the  variations  of  the  elements  of  the  orbit,  ho  regards 
these  elements  as  constant  and  finds  what  may  be  termed  the  perturbation  of  the 
time.  The  publication  of  his  work,  in  which  this  new  mode  of  treatment  is  given, 
entitled  A.useinandersetzung  einer  zweckmassigen  Methods  zur  Berechnung  der  absoluten 

A.  P.  s. — VOL.  xrx.  A 


6  A    NEW    METHOD    OF    DETERMINING 

Storungen  der  Ideinen  Planeten,  undoubtedly  marks  a  great  advance  in  the  determina- 
tion of  the  general  perturbations  of  the  heavenly  bodies. 

The  value  of  the  work  is  greatly  enhanced  by  an  application  of  the  method  to  a 
numerical  example  in  which  are  given  the  perturbations  of  Egeria  produced  by  the 
action  of  Jupiter,  Mars,  and  Saturn.  And  yet,  notwithstanding  the  many  exceptional 
features  of  the  work  commending  it  to  attention,  astronomers  seem  to  have  been  de- 
terred by  the  refined  analysis  and  laborious  computations  from  anything  like  a  general 
use  of  the  method ;  and  they  still  adhere  to  the  method  of  special  perturbations  devel- 
oped by  LAGRANGE.  HANSEN  himself  seems  to  have  felt  the  force  of  the  objections 
to  his  method,  since  in  a  posthumous  memoir  published  in  1875,  entitled  Ueber  die 
Stdrungen  der  grossen  Planeten,  insbesondere  des  Jupiter  s,  his  former  positive  views 
relative  to  the  convergence  of  series,  and  the  proper  angles  to  be  used  in  the  argu- 
ments, are  greatly  modified. 

HILL,  in  his  work,  A  New  Theory  of  Jupiter  and  Saturn,  forming  Vol.  IV  of 
the  Astronomical  Papers  of  the  American  Ephemeris,  has  employed  HANSEN'S 
method  in  a  modified  form.  In  this  work  the  author  has  given  formulae  and  devel- 
opments of  great  utility  when  applied  to  calculations  relating  to  the  minor  planets,  and 
free  use  has  been  made  of  them  in  the  present  treatise.  With  respect  to  modifica- 
tions in  HANSEN'S  original  method  made  by  that  author  himself,  by  HILL  and  others, 
it  is  to  be  noted  that  they  have  been  made  mainly,  if  not  entirely,  with  reference  to 
their  employment  in  finding  the  general  perturbations  of  the  major  planets. 

The  first  use  made  of  the  method  here  given  was  for  the  purpose  of  comparing  the 
values  of  the  reciprocal  of  the  distance  and  its  odd  powers  as  determined  by  the  pro- 
cess of  this  paper,  with  the  same  quantities  as  derived  according  to  HANSEN'S 
method.  Upon  comparison  of  the  results  it  was  found  that  the  agreement  was  prac- 
tically complete.  To  illustrate  the  application  of  his  formula,  HANSEN  used  Egeria 
whose  eccentricity  is  comparatively  small,  being  about  -£%.  The  planet  first  chosen 
to  test  the  method  of  this  paper  has  an  eccentricity  of  nearly  ^.  And  although 
the  eccentricity  in  the  latter  planet  was  considerably  larger,  the  convergence  of  the 
series  in  both  methods  was  practically  the  same.  It  was  then  decided  to  test  the 
adaptability  of  the  method  to  the  remaining  steps  of  the  problem,  and  the  result  of  the 
work  has  been  the  preparation  of  the  present  paper. 

HANSEN  first  expresses  the  odd  powers  of  the  reciprocal  of  the  distance  between 
the  planets  in  series  in  which  the  angles  employed  are  both  eccentric  anomalies.  He 
then  transforms  the  series  into  others  in  which  one  of  the  angles  is  the  mean  anomaly 
of  the  disturbing  body.  He  makes  still  another  transformation  of  his  series  so  as  to 
be  able  to  integrate  them. 


THE   GENERAL   PERTURBATIONS   OF   THE   MINOR  PLANETS.  7 

In  the  method  of  this  paper  we  at  first  employ  the  mean  anomaly  of  the  dis- 
turbed and  the  eccentric  anomaly  of  the  disturbing  body,  and  as  soon  as  we  have  the 
expressions  for  the  odd  powers  of  the  .reciprocal  of  the  distance  between  the  bodies, 
we  make  one  transformation  so  as  to  have  the  mean  anomalies  of  both  planets  in  the 
arguments.  These  angles  are  retained  unchanged  throughout  the  subsequent  work, 
enabling  us  to  perform  integration  at  any  stage  of  the  work. 

In  the  expressions  for  the  odd  powers  of  the  reciprocal  of  the  distance  we  have, 
in  the  present  method,  the  La  Place  coefficients  entering  as  factors  in  the  coefficients 
of  the  various  arguments.  These  coefficients  have  been  tabulated  by  RUNKLE  in  a 
work  published  by  the  SMITHSONIAN  INSTITUTION  entitled  New  Tables  for  Determin- 
ing the  Values  of  the  Coefficients  m  the  Perturbative  Function  of  Planetary  Motion  j 
and  hence  the  work  relating  to  the  determination  of  the  expressions  for  the  odd  powers 
of  the  reciprocal  of  the  distance  is  rendered  comparatively  short  and  simple. 

In  the  expression  for  A2,  the  square  of  the  distance,  the  true  anomaly  is  involved 
In  the  analysis  we  use  the  equivalent  functions  of  the  eccentric  anomaly  for  those  of 
the  true  anomaly,  and  when  making  the  numerical  computations  we  cause  the  eccentric 
anomaly  of  the  disturbed  body  to  disappear.  This  is  accomplished  by  dividing  the 
circumference  into  a  certain  number  of  equal  parts  relative  to  the  mean  anomaly  and 
employing  for  the  eccentric  anomaly  its  numerical  values  corresponding  to  the  various 
values  of  the  mean  anomaly. 

Having  the  expressions  for  the  odd  powers  of  the  reciprocal  of  the  distance  in 
series  in  which  the  angles  are  the  mean  anomaly  of  the  disturbed  body  and  the 
eccentric  anomaly  of  the  disturbing  body,  we  derive,  in  Chapter  II,  expressions  for 
the  J  or  Besselian  functions  needed  in  transforming  the  series  found  into  others  in 
which  both  the  angles  will  be  mean  anomalies. 

In  Chapter  III  expressions  for  the  determination  of  the  perturbing  function  and 
the  perturbing  forces  are  given.  Instead  of  using  the  force  involving  the  true  anom- 
aly we  employ  the  one  involving  the  mean  anomaly.  The  disturbing  forces  employed 
are  those  in  the  direction  of  the  disturbed  radius- vector,  in  the  direction  perpendicular 
to  this  radius-vector,  and  in  the  direction  perpendicular  to  the  plane  of  the  orbit. 

Having  the  forces  we  then  find  the  function   W  by  integrating  the  expression 


dW  A         dQ     .     D 

—f-  =  A.ar-    -  B  .  ar 

n  .  dt  dg 


in  which  A,  and  B  are  factors  easily  determined. 


8  A    NEW    METHOD    OF    DETERMINING 

From  the  value  of  W  we  derive  that  of  W  by  simple  mechanical  processes,  and 
then  the  perturbations  of  the  mean  anomaly  and  of  the  radius-vector  are  found  from 


n 


.  te  =  nfw.dt 

C<tf 

I  — 
J   dy 

being  a  particular  form  for  g. 

The  perturbation  of  the  latitude  is  given  by  integrating  the  equation 

u 

~~ 


COS* 


C  being  a  factor  found  in  the  same  manner  that  A  and  B  were. 

It  will  be  noticed  that  in  finding  the  value  of  n  .  &z  two  integrations  are  needed  ; 
in  finding  the  perturbation  of  the  latitude  only  one  is  required. 

The  arbitrary  constants  introduced  by  these  integrations  are  so  determined  that 
the  perturbations  become  zero  for  the  epoch  of  the  elements. 

In  all  the  applications  of  the  method  of  this  paper  to  different  planets  the  circum- 
ference has  been  divided  into  sixteen  parts,  and  the  convergence  of  the  different  series 
is  all  that  can  be  desired.  In  computing  the  perturbations  of  those  of  the  minor 
planets  whose  eccentricities  and  inclinations  are  quite  large,  it  may  be  necessary  to 
divide  the  circumference  into  a  larger  number  of  parts.  In  exceptional  cases,  such  as 
for  Pallas,  it  may  be  necessary  to  divide  the  circumference  into  thirty-two  parts. 

In  the  different  chapters  of  this  paper  the  writer  has  given  all  that  he  conceives 
necessary  for  a  full  understanding  of  all  the  processe  s  as  they  are  in  turn  applied 
And  he  thinks  there  is  nothing  in  the  method  here  presented  to  deter  any  one  with 
fair  mathematical  equipment  from  obtaining  a  clear  idea  of  the  means  by  which  astron- 
omers have  been  enabled  to  attain  to  their  present  knowledge  of  the  motions  of  the 
heavenly  bodies.  The  object  always  kept  in  mind  has  been  to  have  at  hand,  in  conve- 
nient form  for  reference  and  for  application,  the  whole  subject  as  it  has  been  treated  by 
HANSEN  and  others.  Thus  in  connection  with  HAN  SEN'S  derivation  of  the  function 
TF,  to  obtain  clearer  conceptions  of  some  matters  presented,  the  method  of  BBUNNOW 
for  obtaining  the  same  function  has  also  been  given.  In  some  stages  of  the  work 
where  the  experience  of  the  writer  has  shown  the  need  of  particular  care  the  work  is 


THE    GENERAL    PERTURBATIONS    OF    THE   MINOR   PLACETS.  9 

given  with  some  detail.  And  while  the  writer  is  fully  aware  that  here  he  may  have 
exposed  himself  to  criticism,  it  will  suffice  to  state  that  he  has  not  had  in  mind  those 
competent  of  doing  better,  but  rather  the  large  class  of  persons  that  seems  to  have 
been  deterred  thus  far,  by  imposing  and  formidable-looking  formulae,  from  becoming 
acquainted  with  the  means  and  methods  of  theoretical  astronomy.  In  the  present 
state  of  the  science  there  is  greatly  needed  a  large  body  of  computers  and  investiga- 
tors, so  as  to  secure  a  fair  degree  of  mastery  over  the  constantly  growing  material. 

The  numerical  example  presented  with  the  theory  for  the  purpose  of  illustrating 
the  new  method  will  be  found  to  cover  a  large  part  of  the  treatise.  The  example  is 
designed  to  make  evident  the  main  steps  and  stages  of  the  work,  especially  where 
these  are  left  in  any  obscurity  by  the  formula  themselves.  As  a  rule,  the  formula  are 
given  immediately  in  connection  with  their  application  and  not  merely  by  reference. 
It  has  been  the  wish  to  make  this  part  of  the  treatise  helpful  to  all  who  desire  to 
exercise  themselves  in  this  field,  and  especially  to  those  who  desire  to  equip  themselves 
for  performing  similar  work. 

The  time  required  to  determine  the  perturbations  of  a  planet  according  to  the 
method  here  given  is  believed  to  be  very  much  less  than  that  required  by  the  unmodi- 
fied method  of  HANSEL.  Nearly  all  the  time  consumed  in  making  the  transforma- 
tions by  his  mode  of  proceeding  is  here  saved.  The  coefficients  b(i)  are  much  more 
quickly  and  readily  found  by  making  use  of  the  tables  prepared  by  RUNKLE,  giving 
the  values  of  these  quantities.  Doubtless  experience  will  suggest  still  shorter  pro- 
cesses than  some  of  those  here  given  and  thus  bring  the  subject  within  narrower  limits 
in  respect  to  the  time  required.  If  we  compare  the  time  demanded  for  the  computa- 
tion of  the  perturbations  of  the  first  order,  with  respect  to  the  mass,  produced  by 
Jupiter,  with  the  time  needed  to  correct  the  elements  after  a  dozen  or  more  oppositions 
of  the  planet,  computing  three  theoretical  positions  for  each  opposition,  it  is  believed 
there  will  not  be  much  difference,  if  any,  in  favor  of  the  latter. 

Again,  when  we  wish  to  find  only  the  perturbations  of  the  first  order,  experience 
will  show  where  many  abridgments  may  safely  be  made.  And  whenever  the  positions 
of  these  bodies  are  made  to  depend  upon  those  of  comparison  stars  whose  places  are 
often  not  well  determined,  it  will  be  found  that  the  quality  of  the  observed  data 
does  not  justify  refinements  of  calculation. 

One  of  the  things  most  needed  in  the  theory  of  the  motions  of  the  minor  planets 
is  a  general  analytical  expression  for  the  perturbing  function  which  may  be  applicable 
to  all  these  small  bodies.  Thus  if  we  had  given  the  value  of  all  in  terms  of  a  periodic 
series,  with  literal  coefficients  and  with  the  mean  anomalies  of  the  planets  as  the  argu- 

A.  P.  S. VOL.  XIX.  B. 


10  A   NEW   METHOD    OF    DETERMINING 

merits,  we  would  at  once  have  a  ,f    by  differentiation.     And  since 


only  two  multiplications  would  be  needed  in  finding  the  value  of          ,  whose  expres- 

sion has  been  given  above. 

In  the  present  paper  we  have  dealt  only  with  the  perturbations  of  the  first  order 
with  respect  to  the  mass.  The  method  has  been  employed  in  determining  those  of  the 
second  order  also  for  two  of  the  minor  planets  ;  but  as  those  of  Althaea,  the  planet  em- 
ployed in  our  example,  have  not  yet  been  found,  it  was  thought  best  not  to  give  any- 
thing on  the  subject  of  the  perturbations  of  the  second  order,  until  the  perturbations  of 
this  order,  in  case  of  this  body,  are  known. 

The  writer  desires  here  to  record  his  obligations  to  Prof.  Edgar  Frisby,  of  the 
U.  S.  Naval  Observatory,  Washington,  D.  C.,  and  to  Prof.  George  C.  Comstock, 
Director  of  the  Washburne  Observatory,  Madison,  Wis.,  for  kindly  furnishing  him 
with  observations  of  planets  that  had  not  recently  been  observed;  to  Mr.  Cleveland 
Keith,  Assistant  in  the  office  of  the  American  Ephemeris,  for  most  valuable  assistance 
in  securing  copies  of  observed  places.  And  to  Prof.  Monroe  B.  Snyder,  Director  of 
the  Central  High  School  Observatory,  Philadelphia,  he  is  under  special  obligations  for 
the  interest  manifested  in  the  publication  of  this  work,  and  for  continued  aid  and  most 
valuable  suggestions  in  getting  the  work  through  the  press. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  11 


CHAPTER  I. 

Development  of  the  Reciprocal  of  the  Distance  Between  the  Planets  and  its  Odd 

Powers  in  Periodic  Series. 

The  action  of  one  body  on  another  under  the  influence  of  the  law  of  gravitation 
is  measured  by  the  mass  divided  by  the  square  of  the  distance.  If  then  A  be  the  dis- 
tance between  any  two  bodies,  this  distance  varying  from  one  instant  to  another,  it 

(1  \  2 
J    in  terms  of  the  time.     If 

r  and  r'  be  the  radii-vectores  of  the  two  bodies,  the  accented   letter  always  referring 
to  the  disturbing  body,  we  -have 

A2  =  r2  +  r'2  —  2rr  H. 
If  we    introduce  the  semi-major  axes  a,  «',  which  are  constants,  and  their  relation 

/ 

a  —  a ,  we  obtain 
a ' 


//  being  the  cosine  of  the  angle  formed  by  the  radii-vectores. 

Let  the  origin  of  angles  be  taken  at  the  ascending  node  of  the  plane  of  the  dis- 
turbed, on  the  plane  of  the  disturbing,  body.  Let  II,  II',  be  the  longitudes  of  the  peri- 
helia measured  from  this  point;  also  let/,/',  be  the  true  anomalies.  The  angle 
formed  by  the  radii-vectores  is  (f  +  IT)  —  (/  +  H) ;  and  the  angles  /  +  IT,/  +  IT, 
being  in  different  planes,  we  have 

H  -  cos  (/  -f  II)  cos  (/  +  IT)  +  cos  /sin  (/  +  II)  sin  (/'  +  IT),          (2) 

I  being  the  mutual  inclination  of  the  two  planes. 

To  find  the  values  of  n,  IT,  7",  let  <I>  be  the  angular  distance  from  the  ascending 
node  of  the  plane  of  the  disturbed  body  on  the  fundamental  plane  to  its  ascending 


12 


A    ]STEW    METHOD    OF    DETERMINING 


node  on  the  plane  of  the  disturbing  body.  Let  ^  be  the  angular  distance  from  ascend- 
ing node  of  the  plane  of  the  disturbing  body  on  the  fundamental  plane  to  the  same 
point. 

If  TT,  TI',  are  the  longitudes  of  the  perihelia, 

8,  &',  the  longitudes  of  the  ascending  nodes  on  the  fundamental  plane  adopted, 
which  is  generally  that  of  the  ecliptic,,  we  have 


IT  =  n  -  -  Q'  -  -     . 


(3) 


The  angles  4>,  41?  &  —  &',  are  the  sides  of  a  spherical  triangle,  lying  opposite  the 
angles  t\  180  —  «,  7, 

^  'i',  being  the  inclination  of  disturbed  and  disturbing  body  on  the  fundamental 
plane. 

The  angles  /,  4>,  ^,  are  found  from  the  equations 


sin  |  /sin  \  (^  +  <£)  ~  sin  |  (&  --  8')  sin  |  (i  +  «*) 
sin  |  /cos  |  (^  +  4>)  =  cos  J  (8  --  Q>')  sin  |  (z  --  i') 
cos  |  /sin  |  (^  —  4>)  =  sin  \  (Q  —  Q')  cos  \  (i  +  i) 
cos  J  /cos  I  (^  -  -  $)  =.  cos  |  (8  --  Q>')  cos  |  (^  --  z7) 


In  using  these  equations  when  Q  is  less  than  Qf  we  must  take  \  (360°  +  Q,  —  &') 
instead  of  \  (Q  -  -  Q'). 

We  have  a  check  on  the  values  of  /,  <J>,  ^  by  using  the  equations  given  in  HAX- 
SEN'S  posthumous  memoir,  p.  276. 


Thus  we  have 


cos  p  .  sin  q 
cos  p .  cos  q 
cos  p .  sin  r 
cos  p .  cos  r 
sin  jp 

sin  /sin  4> 
sin  /  cos  4> 
sin  /  sin  (^ 
sin  /cos  (^ 
cos  1 


sin  ^  .  cos  (Q,  —  Q,') 

COS  l' 

cos  i  .  sin  (8  —  Q') 

cos  (8  —  &') 

sin  ^'     sin  (8  —  Q') 

sin  ;j 

cos  p .    sin  (^  -  -  q) 

sin  .;;  .  cos  (i  --  q} 

sin  (^  —  q) 

cos  jj  .  cos  (i  -  -  q} 


(5) 


THE  GENERAL  PERTURBATIONS  OP  THE  MINOR  PLANETS.  33 

To  develop  the  expression  for  (-V  we  put 

cos  /.  sin  IT  =  k  sin  K,  sin  II'  —  kL  sin  K^  ) 

cos  n'  =  k  cos  Kj  cos  /cos  II'  =  kL  cos  .STj,  j 


and  hence 


II  =  cos/".  cos/"  .  &  cos  (n  -  -  K)  -\-  cos  /'.  sin  f  .  kL  sin  (II  —  K^ 
-  sin/*.  cos/'  .  k  sin  (II  -  -  ^T)  -}-  sin/,  sin/'  .  A?i  cos  (II  —  JSQ. 


Introducing  the  eccentric  anomaly  F,  we  have 

cos/"  zz  -  (cos  £  —  e),  sin/1  =  -  .  cos  <p  .  sine, 

e  being  the  eccentricity,  and  $  the  angle  of  eccentricity  ;  and  find 

T        T1 

-  ...  //=  cos  c  .  cos  E'  .  k  cos  (n  —  K]  —  cos  F'  .  ek  cos  (n  —  K] 
a  a 

-  cos  £  .  t'k  cos  (n  —  K)  +  ee'k  cos  (II  —  K) 

+  cos  F  .  sin  F'  .  cos  <?>'  .  fa  sin  (n  —  K^  —  sin  e'  .  e  .  cos  <p'  .  fa  sin  (II  — 

-  sin  e  .  cos  F'  .  cos  <?>  .  fc  sin  (II  —  .5T)  +  sin  £  .e'  .  cos  <|>  .  k  sin  (n  —  J5T) 
+  sin  £  .  sin  t'  .  cos  $  .  cos  $'  .  ^  cos  (II  —  jffi). 


Substituting  the  value  of  r  .  7,  .  //  in  the  expression  for  f-J    we  have 

/  j\  - 

(    J   =  1  +  a2  —  2e  .  cos  F  +  e1  cos  2e  —  2  aee'k  cos  (II  —  /iT) 

+  2ae'k  cos  (II  —  K)  cos  e  —  2ae'  cos  <p  .  ^  sin  (n  —  K)  sin  F 

-  [2aV  --  2a<?&  cos  (n—  K)  +  2flfc  cos  (II  —  7T)  cos  e 

-  2a  cos  ^  .  k  sin  (II  —  JT)  sin  F]  .  cos  t' 

-  2ae  cos  ^)'  .  fa  sin  (II  —  JQ  +  2a  cos  ^)  cos  q>  .  k^  cos  (II  —  JSTj)  sin  F 
+  2a  cos  <^'  .  A;,  sin  (II  —  Id)  cos  e]  .  sin  f' 


a26/2.COS  V. 


Putting  yj,  /^0,  y2,  for  the  coefficients  of  cos  f',  sin  t',  cos  V,  respectively,  and  y0  for 
the  term  not  affected  b}^  cos  t'  or  sin  e',  we  have  the  abbreviated  form 

—  y0  —  y]  .  cos  F'  —  ft0  .  sin  F  -f-  7^  •  cos  V.  (7) 


14  A   NEW    METHOD   OF   DETERMINING 

/  A  \  - 

In  this  expression  for  (  J  ,  yc,  y1?  and  (30  are  functions  of  the  eccentric  anomaly 
of  the  disturbed  body ;  y->  is  a  constant  and  of  the  order  of  the  square  of  the  eccen- 
tricity of  the  disturbing  body. 

In  the  method  here  followed  the  circumference  in  case  of  the  disturbed  body  will 
be  divided  into  a  certain  number  of  equal  parts  with  respect  to  the  mean  anomaly,  g. 

mu            •                           f          -11  4.1         u     no    360°    0  360°    o  360°                              -,  360° 
The  various  values  01  q  will  then  be  (Jr. ,  z. ,  d. ,  .  .  .  .  n  —  1. 

n    7         'it    '         n    '  n 

For  each  numerical  value  of  #,  the  corresponding  value  of  F  is  found  from 

g  —  £  —  e  sin  F. 

Before  substituting  the  numerical  values  of  cos  F,  sin  F,  for  the  n  divisions  of  the  cir- 
cumference, the  expressions  for  y^  yl7  /30,  will  be  put  in  a  form  most  convenient  for 
computation. 
Let 

p.  sin  P  -  2a2  e-  —  2aJc  cos  (n  —K)  \ 

e  f  ($) 

p.  cos  P  =:  2a  cos  <£>'  JcL  sin  (II  —  J5Q,  J 

and 

fr=/'cosp- 1  (9) 

we  find 

($Q  r=/sin  F  •=.  2a .  cos  <p .  cos  <£>'.  &x  cos  (II  —  JKi).  sin  F  +  p  cos  P.  cos  F  —  ep .  cos  P 

yi  =fcosF=  (Z'j?  *  — p  sin  PJ.  cos  F  —  2a .  cos  <p .  k  sin  (n  --  K) .  sin  £  -\-  ep .  sin  P. 

And  from  these  equations  we  find,  since 

/.  sin  (F—  P)  —f.  sin  Fcos  P  —/cos  F.  sin  P 
/.  cos  (F~  P)  r=/cos  F.  cos  P  +  /sin  P.  sin  P, 

/.  sin  (P —  P)  rr  [2a, .  cos  $  .  cos  <p' .  ^  cos  (II  —  JT,).  cos  P 

+2a.cos  (?>.&  sin(Il— JT).  sin  PJ .  sin  F  +    p  —  2ot2  -  sin  P  ] .  cos  F — ep 

f.  cos  (F—  P)  =  [2a .  cos  ^ .  cos  <£' .  ^  cos  (11  -  -  K{) .  sin  P 

—  2a  .  cos  <p .  ^  sin  (FT — K] .  cos  P]  .  sin  F  -f  2a2 .  -  .  cos  P .  cos  F 

e 


THE   GENERAL   PERTURBATIONS   OF    THE   MIISTOR    PLANETS. 

If  we  now  put 

v  sin   V  rz  2a  .  cos  <p  .  Jc  sin  (II  —  K] 

v  cos  V  zz  2a .  cos  ^) .  cos  fy' .  ~k±  cos  (II  —  /iQ 

iv  sin  TF  =  p  —  2a2 .  - .  sin  P 

e 

10  cos  W  —  v  .  cos  ( "F" —  P) 
«?!  sin  JFi=:  v  .  sin  ( V —  P} 
^ul  cos  Wi=  2a2 .  *  .  cos  P, 


(10) 


we  get 


Further,  if  we  put 


/.  sin  (F—  P)  —  w.  sin  (s 

f.  COS  (jF—  P)  =  M!  .  COS  (e  +   TTi). 


=  1  +  a2  —  2a2 .  e'2, 


we  have 


=  j    .  —    e  .  cos 


e'y  i 


(11) 


(12) 


or, 


—  R  —  2e .  cos  F  +  e2 .  cos  2e  +  e'  ./cos 


(13) 


AVe  find  the  value  of  y.2  from 


The  constants,  ^,  JT,  &b  1^,  ;^,  P,  n^,  IF",  w,,  TT^,  ^,  are  found,  once  for  all,  from 
the  equations  given  above.  For  every  value  of  e  we  have  the  corresponding  value  of 
/and  Pfrom  equations  (11) ;  hence,  also  the  values  of  _/sin  ^,/cos  F,  which  are  the 
values  of  /#„  and  ylt  Equation  (13)  furnishes  the  value  of  y0  by  substituting  in  it  the 
various  numerical  values  of  e,  as  was  done  for  j3Q  and  y^  The  value  of  the  coefficient 

y2  being  constant,  we  thus  have  given  the  values  of  (-J    for  as  many  points  along 
the  circumference  as  there  are  divisions. 


16  A    NEW    METHOD    OF    DETERMINING 

We  can  put 

f-  ]   —  y0  —  yi  c°s  ?'  —  ft  •  sin  e'  -{-  /2  .  cos  .  V 

\  u>  / 

in  the  form 

=  [(7-  g.  cos  (*'-#)]  [1-^.  cos  (E'  -§,)], 


in  which  the  factor  1  —  ql  .  cos  (V  -  -  Qi)  differs  little  from  unity.  For  this  purpose,  if 
we  perform  the  operations  indicated  in  the  second  expression,  and  then  compare  the 
coefficients  of  like  terms,  we  find 

y0  =  C+q.q.s'mQ.  sin  Ql 

YL  zz  q  .  cos  Q  -f-  q^  .  C  COS  Q{ 


ft  =  £  .  sin  Q  +  #1  •  <?  sin  Qi 
0=rin-(C+ft)« 

The  last  of  these  equations  is  satisfied  by  putting 

Q,  =  --Q. 
The  remaining  equations  then  take  the  form 


yi  =  to  4-^-0).  cos  Q 
y-2  =  2  .  ffi 

ft  =  (q  —  qi.C).  sin  Q 
The  expressions 

q  .  sin  Q  —  ft  +  £  ] 

g.coBQzry,-,  [ 
5l  .  C.  sin  Q  =  £ 

gL  .  (7.  cos  Q  —  V  J 

satisfy  the  relations   expressed  by  the  second  and  fourth  of  equations  (15),  where 

c=yo  +  <;. 

"We  have  now  to  find  expressions  for  the  small  quantities  £,  q,  f  found  in  these 
equations, 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  17 

Equations  (16)  give 

q.q,.  Csm2Q=  (ft  +  £).£. 
The  equation 

7o=  O—  q.q^sm^Q 
then  becomes 

(yo  +  ?K=(ft  +  £)f  (a) 

From  (16)  we  have,  also, 


from  which,  since  y.2  =  q  .  q^  and  (7=  yQ  +  f,  we  obtain 


and  hence 


Equations  (16)  give  again 

(y,->0f  =  (&  +  £)«?•  (c) 

When  £  is  known,  £  is  found  from  (a)  ;  and  the  difference  between  (a)  and 


gives  >7  when  f  is  known. 

The  equations  (a)  and  (c)  give 


A.  P.  8.  —  VOL.  XIX.  C. 


18  A   NEW   METHOD  OF    DETERMINING 

Deduce  the  values  of  (3Q  +  £,  ^  —  Y\  from  (a)  and  (d),  substitute  them  in  (c),  we  find 


The  last  equation  then  takes  the  form 


This  equation  furnishes  the  value  off;  and  with  f  known,  we  find  £,  >?,  from  equations 
already  given.     The  three  equations  giving  the  values  of  the  quantities  sought  are 


=OJ 


Finding  the  values  of  f,  £,  >7,  from  these  equations,  and  arranging  with  respect  to 
preserving  only  the  first  power,  we  have 


/2 


(9) 


Substituting  these  values  in  equations  (16),  they  become 


h  C  sin  0  •=. 


•n 


(17) 


noting  that  C  =  y0  +  f . 

If  more  accurate  values  of  f,  £,  >?,  are  needed  than  those  given  by  equations  (#), 
we  proceed  as  follows  : 

Substitute  the  value  of  f  given  by  (</)  in  the  second  term  of  the  first  of  equa- 
tions (/'),  we  find,  up  to  terms  including  y22, 


—  4. 


(18) 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  19 

The  last  two  of  (/)  give  also 

c  '£       °*  •  £* 


„  _ 


Introducing  the  values  of/,  F,  given  by  (11),  putting 


(19) 

we  have 

f  =  %.sin^ 
so  that 

<7=yo  +  £.sin2.F.  (20) 

Moreover,  since 

72  —  ?  =  2'.  cos  IF, 
we  find  from  the  expressions  for  f  ,  77,  given  above, 

&  +  £=/.£'.  sin  JP,  . 

7i~>7  =/•>?'.  COS  ^ 

if 


Substituting  these  in  the  expressions  for   q  sin  §,  q  cos  Q,  they  become 

2  sin  Q=f.£  .sin^7 


(22) 


20  A   NEW    METHOD    OF    DETERMINING 

The  value  of  q^  is  found  from 

?L  =  r*  (23) 

The  quantities  q,  q^  Q  can  be  expressed  in  another  manner.     The  equations  (22) 
give 

<f  —f2 .  f2 .  sin  '2F+f2 .  vf2 .  cos  -F; 
from  which  we  derive 


Q=F  +  r=     .  sin  2F  +  |      -         -  sin  4.P+  etc. 

*    ~t~  v  v^n~V'' 

log.  gr  =  log./+  1  log.  (f-  .  sin  2.F+  ^  cos  2^). 


Since  %2  and  ^^  agree  up  to  terms  of  the  third  order,  the  equations  for  £'  and  >/ 
give 

^-V  -  g  (y  +  /)  . 


or 


Further 

£'2  sin  *F-\-  >?'2  cos  2F  =.  1  +  2  -^  (% .  sin  2F —  ^'  cos  ~F)  - 

and 

J  log.  (f2  sin  2^+  >/2  cos  2^)  =  ^  (%  sin  2^—  ^'  cos 

—  TT  (z  s^11 2^  —  %'  cos 
Substituting  the  values  of  £,  ^',  (7,  given  before,  we  find 


(K  sin  ^-  ^  cos  *F)  =  JL  +  -      -   ?      +          cos 


THE   GENERAL   PERTURBATIONS   OF    THE   MINOR   PLANETS.  21 

The  equation  y.,  nz  q .  q{  gives 

l°g-  72  =  log.  q  +  log.  ^ 
Putting: 


log.  ff  =  log. 
we  have  for  3^ 

log.  g1  =  k>g.       —  y. 


Writing  s  for  the  number  of  seconds  in  the  radius,  and  ^0  for  the  modulus  of 
the  common  system  of  logarithms,  we  find 


(24) 


in  which 


log.  q  =\og.f+y 
log.  0i  = 


x  =  s  .    sn  s  —  -        sn 

(25) 
cos  2^-X0  -  --2   cos 


And  for  C  we  have  from  the  first  of  (15) 

C^yo  +  ^.sin2^.  (26) 

By  means  of  the  last  three  equations  we  are  enabled  to  find  the  values  of 
ft  <??  <?i>  £>  w^h  the  greatest  accuracy.  The  equations  (17),  where  not  sufficiently 
approximate,  will,  nevertheless,  furnish  a  good  check  on  the  values  of  these  quantities. 

(A\  2 
J   are  thus  known;    and  substituting  their 

values  corresponding  to  the  various  values  of  g,  we  have  the  values  of  (~\    for  the 
different  points  of  the  circumference. 


22 


A   NEW   METHOD   OF    DETERMINING 


Using  the  values  of  (7,  </,  </b  Q,  just  found,  HILL,  in  his  New  Theory  of  Jupiter 
and  Saturn,  has  given  another  expression  for  (-}  which  we  shall  employ. 
To  transform 


=  (<7—  q  .  cos  (V  -  Q)  )(!-&.  cos  (.'  +  Q)) 


into  the  required  form  we  put 


(27) 


=  seel  £.  seel  fr 
V      (7  ~ 


Then 


(  V  z=  C  [l  —  sin  x  -  cos  (e' —  §)]  [l  —  sin  £1  •  cos  (e  + 

C  [sec2 1%  (1  —  sin  %  .  cos  (e'  -  -  (2))]  [sec2 1;^  (1  —  sin  ^  .  cos  (e'  + 

sec2  |ri 


sec 


C 


cos 


sec 


sec 


Substituting  the  values  of  a,  &,  JV^  we  get 


-  Nn  [1  +  a2  —  2a  cos  (V  —  §)]"  2  [l  +  62  —  2&  cos  (V  + 


(28) 


We  compute  the  values  of  a,  &,  JVJ  corresponding  to  the  different  values  of  g,  and 
check  by  finding  the  sums  of  the  odd  and  the  even  orders,  which  should  be  nearly  the 
same.  If  we  put 

[1  +  a2  —  2a  cos  (E' —  Q)]~s  -  [J  J°>  +  &(1) .  cos  0  +  6(2) .  cos  29  +  &("  .  cos  3,9  +  etc.] 
[1  +  Z>2  —  26  cos  (e'  +  Q)]~s  -  [l  ^(0)  +  BV  .  cos  (e'+  Q)  +  B-\  cos  2  (/  +  §)  +  etc.] 
where  s  =  ^,  0  =:  e'  —  §,  we  are  enabled  to  make  use  of  coefficients  already  known. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  23 

For  2  .  cos  0,  write  x  +  -  ,          and  then  we  have 

7 


+  a2  —  2a  cos  0]~s  =  [l  +  a2—  a  (x  +  ^ 


Expanding  we  have 


,    I    r+l    S_+J    S  +  3    S  +  4       55 

1*2  3         ~T~     ~5~  *  C> 

_  a~|~s  _  -j,    s     a     .    s     s  +  1     a2    ,     s^    s  +  1    s  -f-  2    a3    ,    s     8  +  1    s+2    g  +  3    a* 
^J  1  *  *  ~"  1  *      2      "  ^2  "t"  I  '      2      '      3      '  ^3    '    1  ' 


__ 
2      *      3      ' 


,s     s+  1    s_+2    s  +  3    s  +  4    a8    .       , 
1"      2  3  4~     ^5~  '^r5"1 


And  hence,  for  their  product,  we  have 


/s    a  +  i    s  +  2v  2   8  +  3  .  a*  +  etc.!  /         l 

"  u  •  ~2      3~;    ^^  J  r  "  * 

-4-  fs     S-±^    «2   ,    /s  \2    s  +  l    s  +  2      4        /«     s  +  l\2  s  +  2    s  +  3      6 
"  Ll  *      2  "  VU    '       2  3  U'~2     ;'~3"~'~~4~-tt 

,     (s     s  +  1    ,  +  2y  8  +  3    «  +  4      8  -l    /  o,     1 

u  •  ~2      3  ;  •  ~4~  •  "T^  •  a      tc>  J  r 

-1-   P       1±_1     S+J     /.S.L/'M2     ?-±l      !+-?     S  +  3     «5 

Ll  '      2  3  VI  /   '       2  3  4 

fs     s+l\2s  +  2s  +  3s  +  4      7| 

+  (r  •  T-)  •    »  •   4   •  -5    a  +  etc- 


+  etc. 

But  aj  +  1  -  =  2  cos  0,      or  +  ^  =  2  .  cos  20,     a?3  +  \  =  2  .  cos  30,      etc., 

1          «•  /  '          /y»*  '  /T1*  *  ' 


24  A   NEW  METHOD   OF    DETERMINING 

and  hence 


^ 

=  2,a[l  +  1^  .«*  +  1,  (•-  +3*  •£•*  +  ?  .Ct1^2) 


s  _)_  2      o    i    s     s+ls  +  2s  +  3      4 


,«     a  +  l  /s  +  2\2  s  +  3    S  +  4      .    ,      t    "I 
I  '  ~2"  V    3"  V  '   ~T~     "5" 


s  +  3    j   ,   .     s+1    S  +  3    S+4      ,  (29) 


_0  81       ,  », 

-^T-~2~'~T~-a  L      -f.     4-.a       r*-  4          5 

s     s  +  1    s  +  2    s  +  3    s  +  4    s  +  5      6          , 
"r-"^"-~3^'~"4"'       :,  6  C'' 


and  generally 


>=  2  .  -  .  ' 


....     .  .  .          .  .._ 

Since  s  =  ^,  we  find  from  these  expressions  the  values  of  the  6{0  coefficients  for 

different  values  of  n. 

RUNKLB  has  tabulated  the  values  of  5(?)  in  a  paper  published  by  the  SMITHSONIAN 
INSTITUTION.     Thus  the  value  of 


[1  +  «2  —  2a  cos  (s'—  §)]  ~ 
is  obtained  with  great  facility. 

_  n 

The  value  of  [1  +  62  —  26  cos  (e  +  Q)~]~2  is  found  in  the  same  way. 

We  now  let 

c">  =  i.^r.^.cos2^| 

s(i)=i.JSr.£(i).sm2.iQ) 

And  hence  have 

cf0)  =  i  .  N.  .Z?(0) 


etc.=  etc. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


25 


Multiplying  the  series  [|  6(0)  +  &(1).  cos  0  +  Z>(2).  cos  20  +  6(3).  cos  30  +  etc.] 

by    [J  £(0}  +  B(Y>  cos  (e'  +  Q)  +  ^(2).  cos  2(e'  +  Q)  +  etc.] , 

noting  that  0  —  (&  —  e ',•  and  arranging  the  terms  with  respect  to     cos  *0,    sin  iQ, 


we  find 


=          .  c 


(0) 


.  c 


(2) 


.  c 


(0) 


etc. 


cos 
gin 


sin  20 
cos  30 
sin  30 


etc. 


(31) 


cL  cos  /£  =  &(i).  c(0) 


sn 


^    Cr'          . 

C      *    ' 


and  we  find 


(2) 


(a  j  =  A:t  [cos  £J  .  cos  iO  +  sin  ^  .  sin  . 

=  Tci  cos  (*0  —  JSTJ  =,  ki  .  cos  (i  Q  — 


—  Kt). 


Subtracting  and  adding  the  angle  igt  this  becomes 


-}  i-  &,cos  \i(0 — Q}  —  KI  +  (iq  —  i 

A     I  \      4^  «7   /  \     »/ 

^z  ^  cos   *  ( Q — g]  —  KI  cos .  i (g  —  e')  —  lc>i .  sin  i 
If  we  put 


(32) 


(33) 


—  £i   sin . t (g  —  e')  (34) 


^  K  =  -  ^  „  cos  p  ( QK—gK) 


—  Ki,  J 


(35) 


h  i 


A.  P.  S. — VOL.  XIX.  D. 


26  A   NEW    METHOD    OF    DETERMINING 

n  being  the  number  of  divisions,  we  find 

fa\  lc]  •/  A(S)      •     •/  ,  \  (Qa\ 

(    }  —  AL  K  .  cos  i  (gK  —  E  J  —  AI  K .  sin  ^  (gK  —  e  K)  (ob ) 

\  A) 

If  now,  for  the  purpose  of  multiplying  the  series  together,  we  put 

(0  (0  J&  ^ 

A^-^C^. 
we  have 


.    i>v  ^ 

(«)  (o  («) 

.t.  „  =  2  $,  „  .  cos  y#  +  2  fi/,;  .  sin  v</     J 


=  [2  £[,„  cos  j>#  +  2  Cl>  smvg]  cosi(g—e')—[2  SLv  cos  r#-f-2  $,„  sin  w/]  sh™  (#—£' 


(38) 
Performing  the  operations  indicated  we  get 

(c)  (c)  (c) 

22  cos  («gr  —  is'}.  Cj-,  cos  vg  —     22  J  C<,  „  cos  \_(i-\-  v)  9  —  *V  ]  +22  1  (7(>  cos  [(*  —  i^)  ^  —  te'J 

(s)  («)  (s) 

22  cos  (ig  —  ie').  Ci>v  sin  vg  •=.     22  J  C^>  sin  [(«  +  v)  ^  —  w']  —  22  J  C^>  sin  [(*  —  v)  g  —  tV] 

(c)  (c)  (c) 


—  22  sin  («V?  —  &')  /S',,,  cos  v^r  =.  —  22  J  ^>  sin  [(t-f  v)  ^  —  *V]  —  22  J  ^>  sin  [(*  —  v)  ^r  —  ie'~] 

(s)  (s)  (s) 

—  22  sin  (ig  —  ief)  Si>v  sin  vg  —     22  J  SliV  cos  [(*+  v)  g  —  tV]  —  22  1  /St>  cos  [(*'  —  v)  g  —  u'~\ 


Summing  the  terms  we  find 

(c)  («)  («)  (c) 

=  22  i  (  C,,,  =F  «,  J  cos  [(»  =F  v)  ^  —  &']  =F  i  22  (  a>  ±  «,  J  sin  [(i  =F  v)  g  —  u']  (39) 


(c)          (c) 

From  the  formula  of  mechanical  quadrature  just  given,  we  have  Cit  0,  8it  0,  when 

(c)  (c) 

v  n  0  ;  but  we  know  that  they  are  J  .  CJ,  0,  J  «^  o?  as  shown  by  their  derivation. 
Thus 

(c)  (c)  (c)  (c) 

AI  =  J  Q  o  +  O{  i  cos  (/  +  C/  2  •  cos  2g  +  etc.   I  (c)  (•) 

w  (S)  >  =  2  Cit  v  cos  ^  -f  2  C-,  „  sin  vg 

+  <7U  sin  gr  +  Cit  2  .  sin  2^r  +  etc.  J 

(s)  (c)  (c)  (c) 

At  =  \  BW  +  Ski  cos  ^  +  Si  ,2  cos  2^  +  etc.  ^  (c)  w 

w  (»  J  =  2/Sz>  cos  r^  +  2/S'/>  sin  vg. 

+  /S/;  i  sin  #  +  /Si;  2  sin  2g  +  etc.  J 

Hence  where  v  =  0,  each  series  is  reduced  to  its  first  term. 


THE   GENERAL   PERTURBATIONS    OF    THE  MINOR  PLANETS.  27 

In  the  application  of  the  very  general  formulae  care  must  be  taken  to  note  the 
signification  of  the  various  terms  employed. 


In  case  of 


(c)          2 

«  =  ~  \«  -  cos  P  (ft—flO  —  KI,  J 


n 


(s)          2 

« =  „  &/, « • siu  [« (&—-#*)  —  KI,  J > 


>i 


n  shows  the  number  of  divisions  of  the  circumference ;  and  we  divide  by  ^  m  form- 
ing Jct  K  to  save  division  when  forming  the  coefficients  c,,,  sv. 
The  index  and  multiple  i  shows  the  term  in  the  series 

Ji(0)  +  Z>(1)  cos  (V  —  Q)  -\-  &(2)  •  cos  2(Y  —  Q)  -f  Z>(3) .  cos  3(e'  -  -  Q)  -(-  etc. 

The  double  index  i,  x  shows  the  term  of  the  series  of  La  Place's  coefficients  and 
the  particular  point  in  the  circumference. 

The  index  v  shows   the    general    term  of  the   series  expressing  the   values   of 

(c)  (s) 

AI :IK,  -Af,,,  when  we  give  to  v  values  from  v  r=  0,  to  the  highest  value  of  v  needed  in 
the  approximation. 
2 

In      .&,,„,  ^*(§*  — gK)  —  -5^»  for  each  value  of  i9  there  are  n  values  of  each 

it/ 

quantity. 

(c)          (c)          («)  (c)          («) 

The  next  step  is  to  express  the  w  values  of  A0  ,  ^  ,  Al  ,  u42  ?  A  >  etc.,  respec- 
tively in  terms  of  a  periodic  series.  And  since  these  quantities  are  functions  of  the 
mean  anomaly  #,  if  we  designate  them  generally  by  Y,  of  which  the  special  values  are 

~y  ~y  ~y  ~V~ 

we  have 

Y—  Jc0  +  c,  cos  g  +  c2  cos  2g  +  etc.  )  0 

+  «i  sin  ^  +  s2  sin  2g  -\-  etc.  ) 

The  values  of  c,,  «„,  in  this  series  are  found  from  the  n  special  values  of  Y. 


28  A   NEW    METHOD    OF   DETERMINING 

From 


(C)  (8) 

I  ,  or  AI    —  |  c0  +  ^!  cos  g  +  c2  cos  2</  -f-  etc. 
+  »!  sin  g  +  s2  sin  2#  +  etc., 


and  similarly,  for  every  other  value  of  x  in  Ait  K,  Ait  K,  we  have  a  check  on  the  values  of 
€„,  s,,  in  each  series.  Thus  if  in  case  of  sixteen  divisions  of  the  circumference  we 
take  g  =r  22  ,°  5  and  find  the  value  of  the  series,  the  sum  of  the  terms  must  equal  the 

(c)  (s) 

value  of  At,  «,  Ait  K,  corresponding  to  g  =  22  .°  5.  And  this  check  should  be  employed 
on  each  series,  using  that  value  of  g  that  gives  the  most  values  of  c,  and  sv.  If  i 

nds  to  i  —  9,  we 
In  the  equation 


(e)  («) 

extends  to  i  —  9,  we  have  ten  separate  checks  for  the  values  of  Alt „  Ait  KJ  respectively. 


Y=  \CQ  +  <?! .  cos  g  -{-  c.2.  cos  2g  +  c3 .  cos  3f/  +  etc. 
+  «i  •  sin  g  +  s2 .  sin  2^r  +  s3 .  sin  3</  -f  etc., 

if  the  circumference  is  divided  into  twelve  parts,  each  division  is  30°.     Then  for  the 
special  values  of  Y  we  have 

YQ  =  Jc0  +  ct  +  c2  +  c3  +  etc. 

F!  =  \cQ  +  G!  .  cos  30°  +  c, .  cos  GO6    +  c3  cos  90°    +  etc. 
+  s,   sin  30°  +  s.2  sin  60°    +  ss  sin  90°    +  etc. 

« 

F2  =  Jc0  +  G!  .  cos  60°  +  c.2 .  cos  120°  +  cs  cos  180°  +  etc. 
+  sx    sin  60°  +  s., .  sin  120°  +  sa  sin  180°  +  etc. 

rn  =  Jcb  +  CA  .  330°       +  c2 .  cos  300°  +  c3  cos  270°  +  etc. 
+  s1 .  330°       +  s.2 .  sin  300°  +  s3  sin  270°  +  etc. 


In  the  same  way  we  proceed  for  any  other  number  of  divisions  of  the  circum- 
ference. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  29 

Now  let 

(0.6  )=r0+:r6   ($)  =r0-r6 

(1.7  )=Y1+Y,    (I)  nri-F, 
(2.8)=:F2+:r8  .  •  (|)  =  r2-F8 


=  F5  +  Fu          (A)r=  F6-  Fu 

Then 

3(c0  +  2c6)=:    (0.6)+    (2.  8)  +  (4.  10) 

3(c0-2cc)  =    (1.7)+    (3.  9)  +  (5.  11) 

3(c2  +    c4)  =    (0.6)  —  [(2  .  8)  +  (4  .  10)]  sin  30° 

3(c2-     c4)  =  [(1.7)+  (5.11)]  sin  30°  —  (3.9) 

3(s.2  +   s4)  =  [(1.7)--  (5  .  11)]  cos  30° 

3(,92  -  •  s4)  =  [(2  .  8)  -  -  (4  .  10)]  cos  30° 

3(Cl+    (%)=    (f)  +  [(I)  -(^)]  sin  30° 


6.c3  =    (#)-(«  +  (A) 
3(Sl+   ,6)  =  [(I)  +  (Ty  ]  sin  30°  + 
3(Sl-  85)  =  [(f  )  +  (A)]  cos  30° 


The  values  of  these  coefficients  can  be  easily  verified  by  finding  the  values  of 
each  one  from  the  sum  for  all  the  different  values  of  Y  as  given  in  the  series  for 

V    V    V  V 

-1-  o?  •*  i>  -*•  2>  •  •  •  •   J-  11  • 

When  we  divide  the  circumference  into  sixteen  parts,  each  division  is  22.°  5.  We 
find  the  values  of  Y0)  Y^  Y2,  .  .  .  .  Y^  as  in  the  case  of  twelve  divisions.  To  find 
the  values  of  cv  and  s,,,  in  the  case  of  sixteen  divisions,  we  put 

(0.8  )=Ya+Ys         (f)   =T,-T, 

(i.9)=r,+  r,      (i)  =r,-r, 

(2.10)=  T,+  r10      (A)=  r2-  rIO 

(7  .  15)  =  F,  +  Fu        (  A)  =  F7  -  FB 


30  A   NEW    METHOD   OF   DETERMINING 

(0.4)  =  (0.8)    +(4.12)       (0.2)  =  (0.4)  +  (2.6) 

(1.5)  =  (1.9)    +(5.13)       (1.3)  =  (1.5)  +  (3.  7) 
(2.  6)  ='(2.  10)  +  (6.  14) 
(3.  7)  =  (3.  11)  +  (7.  15). 

Then 


4(c2  +  O      =  (0.8)  -(4.12)  ' 

4fo-  c0)      =  {  [(1  .  9)  -  (5  .  13)]  -  [(3  .  11)  -  (7  .  15)]  j  cos  45 
4(s2  +  s6)      -  \  [(1  .  9)  -  (5  .  13)]  +  [(3  .  11)  -  (7.  15)]  }  cos  45° 
4(s2-s,)      =(2.  10)  -(6.  14) 
8.d=(0.4)  —  (2.6) 

• 

8.  s4  =  (1.5)—  (3.7) 
4(c,  +  c,)     =  (f)  +  [(A)  -  (T64)]  cos  45° 
4(Cl  -  e,)     =  [(i)  -  (A)]  cos  22°  5+  [(&)  -  (A)]  cos  67  °  5 
4(c,  +  c3)     =  (-1)  -  [(A)  -  (A)]  cos  45° 
4(c,  -  c5)     =  [(I)  -  (A)]  sin  22  .«  5  -  [(-ft-)  -  (T%)]  sin  67  .«  5 
4(Sl  +  S7)      =  [(i)  +  (^)  ]  sin  22  .«•  5  +  [(&)  +  (A)]  sin  67  «  5 
4(s,  -  «,)     =  [(T%)  +  (A)]  cos  45°  +  (A) 
4(s3  +  *)      =  [ft)  +  (A)]  cos  22  ."  5  -  [(T«T)  +  (A)]  cos  67  .»  5 

cos  45°-  (A). 


When  the  circumference  is  divided  into  twenty-four  parts,  each  part  is  15°. 
Let 

(0.12)=  ro+  FJ2  (0.6)  =  (0.12)  +  (6.  18)  (4)  =  (0.12)  -(6.  IS) 
(1.13)=  Fx+  F13  (1.7)  =  (1.13)  +  (7.  19)  (*)  =  (1.13)  -(7.  19) 
(2.14)  =  F2+  ri4  (2.8)  =  (2.14)  +  (8.20)  (|)  =  (2  .  14)  -  (8  .  20) 

(11.23)=  Fu+  F23     (5.  11)  =  (5.  17)  +  (11.  23)     (^-)  =  (6  .  17)  -  (11  .  23) 


THE   GENERAL   PERTURBATIONS   OF   THE   MINOR  PLANETS.  31 

Then 

6(c0  +  2  .  c12)  =  (0  .  6)  +  (2  .  8)  +  (4  .  10) 


6(c2  +  cjo)  =  (£)  +  [(I)  -  (^)]  sin  30° 

60,  -  Clo)  =  [(*)--  (A)  ]  cos  30° 

6(c4  +  c.  )  =  (0  .  6)  —  [(2  .  8)  +  (4  .  10)]  sin  30° 

6(ct  —  c8  )  =  [(1  .  7)  +  (6  .  11)]  sin  30°  —  (3.9) 

G)s,  +  «10)  =  [(I)  +  (TST)]  sin  30>  +  (f) 

6(s2-s10)  =  [(*)  +  (A)]  cos  30° 

6(s4  +  ss  )  =  [(I)  —  (^)]  cos  30° 
Q(s,-ss) 


Further,  let 


23 


Then 

6(0,  +  c,, )  =  (A)  +  [(A)  —  (M)]  cos  30°  +  [(A)  —  (A)]  cos  60° 

6(c,  —  c,,)  =  [(A)  —  (H)]  cos  15°+  [(A)  — (A)]  cos45°  +  [(1'V)  — (A)]c°s75c 

6(c3  +  c, )  =  (A)  -  (A)  +  (A) 

6(c3  -  c, )  =  j  (A)  -  (M)  -[(A)  -  (A)]  -  [(A)  -  (A)]  I  cos  45° 

6(cj  +  c,)  =  (A)  —  [(A)  —  (H)]  cos  30°  +  [(A)  —  (A)]  cos  60° 

6(c5-c7)  =  [(A)-(tt)]  sin  15°  -  [(A)  -  (A)]  sin  46»  +  [(A)-(A)]™W 

6(.,  +  «„)  =  [(A)  +  (»)]  sin  15°  +  [(A)  +  (A)]  sin  45°  +  [(A)  +  (A)]  *in  75° 

6(8l  —  «,,)  =  [(A)  +  (if)]  sin  30°  +  [( A)  +  (A)]  sin  60°  +  (A) 

6(«b  +  s,  )  =  {  (A)  +  (tt)  +  (A)  +  (A)  -  [(A)  +  (A)]  }  cos  45° 

6(Si  +  «, )  =  [(A)  +  (H)]  cos  15°  -  [(A)  +  (A)]  cos  45°  +  [(^)  +  (TV)]  cos  75° 
6(s5-s7  )  =  [(A)  +  (if)]  sin  30=  -  [(A)  +  (A)]  sin  60°  +  (A). 


32  A    NEW   METHOD    OF    DETERMINING 

When  the  circumference  is  divided  into  thirty-two  parts,  each  part  is  11°.  25 
Let 

(  0.16)=  r0  +  Yu  (0.8  )  =  (0.16)  +  (  8.24)  (0.4)  =  (0.8  )  +  (4 
(  1.17)=  F,+  r17  (1.9  )  =  (1.17)  +  (  9.25)  (1.5)  =  (1.9  )  +  (5 
(  2.18)=  F2+rM  (2. 10)  =  (2. 18) +  (10.  26)  (2.6)  =  (2.10)  +  (6.14) 


(15.31)=  F15+F31       (7. 15)  =  (7. 23) +  (15. 31)  (0.2)  =  (0.4  )  +  (2.6  ) 

(1.3)  =  (1.5  )  +  (3.7  ) 

—  (  8.24)  (f)  =  (0.8  )  — (4. 

—  (  9 . 25)  Q)  =  (1 . 9  )  —  (5 . 

:  (|)  =  (2. 10) -(6. 14) 

-  (7 . 23)  —  (15 . 31)  (f)  =  (3 . 11)  —  (7 . 15) 

Then 

8(c0+2.c16)  =  (0.2)  +  (1.3) 
8  (c0—  2 .  c16)  =  (0 . 2)  —  (1 . 3) 


8  (c2 -  Cl4)  =  [(i)  -  (A)J  cos  22  .°5  +  [(A)  -  (A) J  cos  67  .°  5 

»-'  \*/4     I       12/  —  \^"/ 

8  (c4  —  c12)  =  [(I)  —  (f )]  cos  45° 

+  c10)  =(|)-[(T2o)-(T64)]cos45° 

-  Clo)  =  [(i)  -(A)]  sin  22  .<>  5  -  [(A)  ~  (A)]  s^  67 .°  5 

16. c8  =(0.4) -(2. 6) 

8 (s2  +  su)  =  [(i)  +  (A)]  sin 22 .° 5  +  [(A)  +  (A)]  sin  67 .° 5 

8(s2-s14)  =  [(A) -(A)]  cos  45°  + (A) 


R  fa    --  o     ^      

O  ^o4          6i2y      — 

8  (s6  —  s10) 


/2\ 
VS/ 

[tt)  +  (A)]  cos 22.°5-[(A)  +  (A)]  cos  67  .'  5 
[(A) -(A)]  cos 45° -(A)- 


TUB  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          33 

Further,  let 


\  5  \  —   V 
3TV  —    *  M 


And  besides,  let 


[(A)  -  (tf  )]  cos  11-.25  +  [(A) 
[(TV)  -  (tf  )]  sin  11°.25  -  [(^) 


cos  78°.75 
sin  78°.75 

[(A)  -  (**)]  cos  22°.5   +  [(JL)  _  (i|)]  cos  67°.5 
A)  -  tt!)]  sin  22°.5   -  [(A)  -  (if)]  sin  67°.5 
(  A)  -  (tf)]  cos  33°.75  +  [(  A)  -  (if  )]  cos  56°.25 
(TS»)  -  ttf)]  sin  33°.75  -  [(,fr)  -  (i|)]  «n  56°.25 
'"  =  (A)  +  [(A)  -  (it)]  cos  45° 
'"  =  (  A)  -  [(A)  -ttf)]  cos  45°  . 

'     =  [(  iV)  +  (if)]  sin  11°.25  +  [(A)  +  (A)]  ^n  78°.75 
=  [(  iV)  +  («)]  cos  11°.25  -  [(A)  +  (A)]  cos  78°.75 

?    =  [(A)  +  («)]  ^  22°.5    +  [(^)  +  (fi)]  sin67°.5 
'  =  [(AM  tt4)]cos22°.5  -[(^)  +  (io)]cos67°.5 

"  =  [(A)  +  (tt)]  sin  33°.75  +  [(^)  +  (J|)]  sin  50°.25 
>"  =  [(A)  +  («)]  cos33°.75-  [(Jj-)  +  (W-)]  cos56°.25 
"'  =  [(A)  +  (41)]  cos  45° 
'"  =  [(A)  +  («)]  cos  45° 


A.  P.  S. — VOL.  XIX.  E. 


34  A  NEW    METHOD   OF   DETERMINING 

Then 

S(Cl  +  Cu)  =  A"  +  A' 

8(Cl—  C*)  =  A  +  A" 

8  (*  +  €„)  =  &"  +  3' 

8  fa  -  c13)  =  [  A  —  A"  +  B  +  JB"]  cos  45 


8  (c3  -  cu)  =  [  J.  -  -4"  -  (-5  +  J?" 
8(c7  +  c9)  =  A"'  —  A 


S(Si  —  8^=C"'+C' 

8  («3  +  *!»)  =  [D  +  D"  ~  (  C  -  C")]  cos  45 


8  (85  +  «„)  =  ID  +  Z>"  +  (7—  C"]  cos  45° 
8(86  —  «„)  =  />'  --D'" 

8(s7  +  s9)  =  D  —  i>" 
8(57  —  s9)  =  —  C""+  C". 

The  expressions  for  the  determination  of  the  values  of  cv  and  sv,  just  given,  are 
found  in  HANSEN'S  Ausewandersetzung,  Band  I,  Seite  159-164. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  35 


CHAPTER  II. 

Derivation  of  the  Expressions  for  BESSEL'S  Functions  for  the  Transformation  of 

Trigonometric  Series. 

The  value  of  Qj    given  thus  far  is  found  expressed  in  a  series  of  terms  the  argu- 

nents  of  which  have  the  eccentric  anomaly  of  the  disturbing  body  as  one  constituent. 
3ut  as  the  mean  anomaly  of  both  bodies  is  to  be  employed,  it  will  be  necessary  to  make 
me  transformation ;  and  the  next  step  will  be  to  develop  the  necessary  formulae  for  this 
mrpose.  HANSEN,  in  his  work  entitled  Eatwickelung  des  Products  einer  Potenz  des 
Radius  Vectors  et  cet.,  has  treated  the  subject  of  transforming  from  one  anomaly  into 
mother  very  fully ;  what  is  here  given  is  based  mainly  on  this  work. 
Calling  c  the  Naperian  base,  and  putting 

ni    f*y  —  1-  fti'    /i«Y  — 1 

y  — c      >          y  —  °       > 

ve  have 


'=  (cos  e  +  V —  1  sin  <0  (cos  e'+'  V —  1  sin  e')> 
ilso 


•,    £,__  _    I      gin      *, 


•=.  cos  (i  e  —  i'  e')  +  V  —  1  sin  (i  e'  —  i'  e'). 

Denoting  the  cosine  and  sine  coefficients  of  the  angles  (is  —  i'  e')  by  («,  i'9  c) 
d  (i,  i',  s)  respectively,  the  series 


F—^^  (i,i,  c)  cos  (ie  —  if  e'  )  —  22  V—  T(^,  i',  s)  sin  (ie  —  i'e1)  (1) 

;an  be  put  in  the  form 


F  =  i  2  2  5  (*;  i',  c)  -  V^  (*,  t%  s)  |  y1  y'1".  (2) 


36  A  NEW   METHOD    OF   DETERMINING 

In  a  similar  manner  we  get 

F  =  J .  2  2  { ((i,  h',  c))  —  V— T(te  A',  8))  y< .  a'-*',  (3) 

where 

z'  =  c-*^. 
We  have  now  to  find  the  relation  between  y  and  z. 

Let 

g  n  the  mean  anomaly, 
and  s  =  the  eccentric  anomaly. 

Then  from 

g  zz  8  —  e  sin  e, 


introducing  V  —  1?  we  get 


g  V —  l  =  eV  —  1  —  esine  V — 
Since 

2  V—  l.sinerry  —  2T1, 
we  find 


from 


7/  —  r€  y—i 

y  —  ° 


we  obtain 


V—  1  =log.«, 
V—  l  =  log.y, 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  37 

and 


(4) 
Thus 


g  V—  1  =  log.  z  —  log. 
and  hence 

z  =  y.c    "2"  .  (5) 

From 

z  =  y .  c~  ^(y-y~'\ 

we  have 
and 


Let  I  be  denoted  by  X ;  then 

&*  -  y~\  (8) 


and 

c^(y~y  )  —  cl  •  y .  c~ il  •  ^  .  (9) 

But 

C-7lA-V  .  C^A  ^rz  Tl  —  ^  .  y    +  T-O   •  2/2  •   -  jVs  '  ^    "^"  1234  '  &*   ^  e^Ct) 


38  A   NEW   METHOD   OF   DETERMINING 

and 

+  &  .  y      +        2  .  f    +  J        .  ys 


Performing  the  operations  indicated,  we  have 


2      liars 


etc 


\  /  _2  . 
O  ^    + 


,     ;'3p  h^s  \  f 

+  1^3  -      I^2^4  db  etc.]  (y  <    - 

F  Ctc0 


,     ^"^    /-i  _          W      _,      _  A^4 
"  1.2..m  V        "  l.w-fl        1.2.ml. 


LC 


__  ^2    ,    JW  .         ^6        ,         ^8 

12.22          p  22  33        I2  22  32.42    ' 

(l'3)3  ,'6)6  ,757  \        / 

+  *  -  i4  +  TO  -  K^ki  ±  etc-)  (»  -  y 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


39 


As  we  may  write  li  in  place  of  i,  we  have,  thus,  also  given  the  value  of  cli(y    V  l)« 
put 

+  00  ( — TO) 


+  GO       (m) 


Then,  from  the  preceding  developments,  we  see  that 


(  —  ni) 


(m) 


m    (m) 


m    (m) 


(m) 


Again 


(-3) 


(1)  (2) 


+  00        (m)  (0)  (1)  (2) 

-oo  e/AA   .  ym   =  Jti  +  '/M  .  2/     +  ^A  .  f 

(-1)  (-2) 


(-3) 


'~3  +  etc. 
/3    -f  etc. 

3    +  etc. 
'~3  +  etc. 


+  co        (—TO) 

Comparing  the  values  of  2_oo  Jl^  .  y~m  and 
have 


7~  7  7,-i  ^3  W  KX'  f          -1 

^tt  :-  Jhx     -Kk--  ^j  +  p-^  —  i^;3^r  ±  etc.,     tor  y    , 


(1)  (1) 

"  " 


(-2)  (2) 


(2)  (2) 


etc.  r=  etc.  — 


7,3)3  1655  L7)7 

ft  AT        |          ft  A  /tA  ..  ^«  -i 

—  -^  -  2-  +  p-23j  —  jTg^g^  dz  etc.,      for  y\ 


+  11   A  -  .  ft  O 

,20204  ^F  etc.,  for  y  \ 

i.£i  1    .  —  .'J  Ji   ,£i  .O.4 

etc.,  for  2/2, 


1.2         P.2.3          12.22.3.4 
etc. 


(10) 


(ii) 


(12) 


(13) 


40  A   NEW   METHOD    OF   DETERMINING 

Comparing  the  values  of  2_*  J,lX  -  ym  and  c7^— 2T1)' 
we  get  the  same  expressions  for  ym  and  y~m. 

(1)  (2) 

We  see  from  the  values  of  «7AA  ,  JhK  ,    etc.,   found     above,    that     the    general 
term  is 

J,WI)TO                 Awi+2  j^wi+2                          ^jWi+4  ^[wi+4 
T        —  _          __  _      I     I      pf"O 

"A       ~  1.2...m         r.2..m.m+l  "    I2.22...m.m-f  l.m+2 


1    1.2.m+l.m+2 

Further,  we  have 


and,  by  putting  m  =  7i  —  it 
this  becomes 

Ji  —    J  n.i  /-i  r\ 

*  —  "h\    •  y  (^j 

Let 

+  00  (h)  ^ 

+  "    (*> 


Multiplying  the  second  of  these  equations  by  z~h.  dg, 
we  obtain 


+  co        (i) 

\z-h.dg  =  2_    Ph  .dg. 


Integrating  between  the  limits  +  n  and  —  7t, 
we  have 


THE   GENERAL  PERTURBATIONS   OF   THE   MINOR  PLANETS.  41 

From 


z  =  cgv  l  =  cos  g  +  V —  1  sii 
we  have 

dz  ==  ( —  sin  g  -f  V —  1  •  cos  g}  dg ; 
also 


z  V —  1  —  V —  1  cos  g  —  sin  g. 
Therefore 


dz  —  z  V— 1.<<7, 
and  (17)  becomes 


In  like  manner  we  find 

'(A)  i         /» 

n.  —  1    .  j 

"  27r1A:rTJ 


Integrating  by  parts  we  have 

™  h 


h-C  y^.z^.dz.  (18) 

O        /        1~  *     '    \J  / 


x-,  . 

Comparing  this  value  of  Qt  with  that  of  Ph,  we  obtain 


or 

(0         ,-         (A) 


s-fc  •  (19> 

A.  P.  8. VOL.  XIX.  F. 


42  A    NEW   METHOD    OF    DETEEMINING 

Thus  we  have,  between  the  mean  and  the  eccentric  anomaly,  the  relations 


In  the  application  of  these  relations,  since 

MO 


the  expression  for  F  is  changed  from 


F  =  J  2  2  {  (;,  f^,rc)  -  V- 1  ft  *\  *)  j  y*.  2/'  - 
into 


F-^^  jfttVc)  —  V—  lft*V)    tf. 

The  other  value  of  JP  is 


((t,  ^',  c))  -  V-  1  ((*,  ^, 

A  comparison  of  these  two  values  gives 

(-*')  $ 

((«,  7^',  c)J  =  2  P_,t,  («,  t',  c)  ±z  2  .  £,  .  J"//v     (*,  ^',  c) 

In  transforming  from  the  series  indicated  by  (i,  i',  c)  into  that  of  ((?,  A',  <?)),  it  is 
evident  that  h'  is  constant  in  each  individual  case,  and  *'  is  the  variable. 

Thus  we  find,  beginning  with  i'  =  ^', 

7/       (h'—hr)  -,,  _  -,       O-(v-i)) 

((i,  h',  c))  =  |  .  «7y  v     (t,  A',  c)  +  ^  .  ^v  0;  *'—  1,  c)  +  etc. 

^'+  1  c    +  etc- 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  43 

To  transform  from  f(t,  £',  e)J  into  (i9  i'9  c) 
we  have 

(i,  »',  c  )  =  2  §_,*  }  ((*,  A',  c))  =  2  Ji,*  *  }  ((*;  h',  c)). 


Here,  i'  is  the  constant,  and  h'  the  variable  ;  and  for  the  different  values  of  7i',  begin 
ning  with  h'  —  i'9 
we  find 

(0)  ((i'-l)-t')) 

(V,  i'9  c)  =  Jf  v  («,  *'  e)j  +  «/(i'-i)  v        ((«,  *''—  1,  c))  +  etc. 


-t-e^+Dv        (0>v  +  l,  c))  +etc. 
The  expression 

^  -  l^rn  (l  '  "  1^+1  +  1.2.m+l.m+2  ~"  1.2.S.m+l,m+2.«i+3  ^  GtC< 

enables  us  to  find  the  value  of  Jhx  for  all  values  of  m. 

A  simpler  method  can  be  obtained  in  the  following  manner : 


,._i\  (0)  (1)  (-1)  (2)  (-2) 

-*     ) 


Putting  ci~y~    in  the  form 

(0) 
=  J       +  J      .  y-J          y+    +J 

fl-^  Fl-%  fl~2  - 

we  have,  for  the  differential  coefficient  relative  to  y, 


(1)  (2)  (-1)  (2) 

,«+2.e7.e.y 

"•"2"  """2" 


If  we  multiply  the  second  member  of  the  first  equation  by  ^|(1  +  y  2)>  we  have 
an  expression  equal  to  the  second  member  of  the  second  expression,  and  by  comparing 


the  two  we  find 

(m+l)  (m-1)   ^.  (m) 


(22) 


44  A   NEW   METHOD    OF    DETERMINING 

Let 

(m) 
h-iT 


-f^>  -  (23) 


then 

(m)  (m-1) 

J  e  —  J  e  '  2 

From  this  general  expression  we  find 

(1)  (0) 

(2)  (1)  (0) 


etc.  zz     etc.     z=     etc. 

(m) 

From  the  values  here  given,  since   — jjj-f-  is  put  equal  to  pm,  we  have,  by  increas- 


ing  m  by  unity, 


(m+l) 
" 


Putting  -7  zz  rw,  equation  (22) 

h-^ 

takes  the  form 

^m  •  Pm+l     I     J-  —  t 

From  this  we  find 


1 

I'm     - 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  45 

We  also  have 


i 

~  - 


(25) 


a  form  more  convenient  in  the  applications. 

(m) 

The  general  expression  for  J  »  is 

2 


(m)  (0) 

e=J,e'Pl-p2.p3..  -Pm  (26) 

-  - 


where 

,(»)  I*   ,      I 


1?  ±  etc" 
if  we  put  l  —  h^. 

From  the  expression 

(—  i')  i         (h'-V) 

((*,  h',  c))  =  2P_A/  (*,  tf,  c)  =  2  ^  Jhw  (i,  i,  c) 

it  is  evident  that  when  &'  z=  0,  or  when  both  f  and  h'  are  zero,  this  expression  cannot 
be  employed. 

To  find  the  values  for  these  exceptional  cases  let  us  resume  the  equation 


/—I 


When  h  =  0  we  have 


(i) 


46  A   NEW   METHOD    OF    DETERMINING 

The  equation 

z  =  y.c-£(y~y~^ 

gives 

dz         dy        e  /-«     ,       _ 2\  j..  /9Q\ 

••  —  ~  —  ^  (.*•  T~  y  )  dy. 

/ 

Hence 

d)  /.i-T^-i 

-»-v  1 


, 

— 1 


jo  is  a  whole  number 


,-"V— 


except  when  jt?  =  1,  when  this  integral  is  27tV — 1- 
Hence  it  follows  that 


When  i  =  0,  we  have 


(i)          (-i) 

PQ     •=.      PQ       ZZ      5 


(0) 

Po    =    1. 


Using  the  expression 


(-i')  (-i')  (-t'-l) 

^  •  /,/   ,,\\  v     p       /,'  «•'   ^,\  —    p       /7*  /'  r"i  _i_  p   ,      /?"  /'_i_  1    r\ 

z,  li ,  cjj  —  2, .  x'—y  ^,  *  >  c;  —  -^ -A'  I,*?  *  > 6^  n^  -L-W    \^  «-  T^  J-j  °y 


we  have 

((o,o,c))_=(o,o,c)-ai'(o,i,c) 

for  the  constant  term,  the  double  value  of  this  term  being  employed. 


THE    GENERAL   PERTURBATIONS   OF    THE   MINOR   PLANETS. 

For  Ji'  —  0,  we  have 

((1,  0,  c))  =  (1,  0,  c)  -  X  (1,  1,  C)  -  x  (1,  -  1,  c) 

((1,  0,  *))  =  (1,  0,  8)  -  V  (1,  1,  8)  -  X  (1,  -  1,  8) 

.  ((2,  0,  o))  =  (2,  0,  c)  -  X'  (2,  1,  c)  -  X'  (2,  -  1,  c) 

.    ((2,  0,  6-))  =  (2,  0,  8)  -  X  (2,  1,  8)  -  X  (2,  -  1,  8) 

etc.  —  etc. 

In  what  precedes  we  have  put 

g  r=  the  mean  anomaly, 
e  i=  the  eccentric  anomaly, 
c  —  the  Naperian  base, 
z  —  cf*-\ 

y  =  tfv~\ 


and  obtain 


where  c~  T&    V  ^  is  expressed  in  a  series,  the  general  term  of  which  is 


.m 


1.2.m+l.m+2         1.2.3.m+l.m4-2.m+3 


Thus 


l.m-j-I         1.2.m-f-l.m-f-2         1.2.3.m+l.m+2.m+3 


±  etc.  )y" 


We  have  also  put 


and  since 


(—m)  (m) 


48 

have  found 


A   NEW    METHOD    OF    DETERMINING 


(m) 


(ft-O 


if 


Again  supposing 


we  have  found 


m  •=.  h  —  i. 


(i) 

p      -fc 

•*    •  * 


(i) 


(A-t) 


Thus  we  have 


(h-i) 


y 


(h-i)  r  —  -  "i 

—  Jh\     |_cos  *£  +  sin  is  V  —  1  J  5 


(A—  i) 


Equating  real  and  imaginary  terms,  we  have 


i       A=00_i    (h~i} 
COS  *V=:-.  2A=_£  Jh\    •  COS 

t-          A=ooJ      (h-i) 

siuie=-.^h^Jh,    .sin 


(29) 


THE    GENERAL   PERTURBATIONS   OF    THE    MINOR    PLANETS. 

We  notice  that 

(i)          (-i) 


For  all  other  values  of  i 


(0) 

Po   =    1. 


(0 

o   =    0. 


49 


If  a  large  number  of  the  J  functions  are  needed  they  are  computed  by  means  of 
equations  (24)  to  (27),  as  shown  in  the  example  given  in  Chapter  V. 
If  we  wish  to  determine  any  of  them  independently  we  have  from 


(«)__        hm)m    r-      ^  fcp 

**  Z       1.2...m  L        "  Lm+^        1.2. 


fr4. 


m+l.m+2       i.2.3.m+l.w+2.m+3 


d  ' 


(0) 


(i) 


(2) 


(3) 


1  —  -    '-  -\ — - .    -  -J-  etc. 

14  4      16         36     64    ' 


7,2     pi          7,4        p*  7,6         pe  ~] 

I         '  I          gf Q 

"  2  '4   "      12  '16      "  144  '64   " 


1.2  3  '  4   ~^~  24  '  16 


T.2.3 


.  .- 

44         40     16 


w        (j,  e-Y  r         /»2  ^  ~\ 

^  =  ra3i4[1--j-f±etcJ 


(30) 


(m) 


In  these  expressions  we  have  written  for  X  its  value  Je. 

Since  h  has  all  values  from  &=+co  to --co  we  find  any  value  of  J"AA  by  at- 
tributing proper  values  to  h. 

From  equations  (29)  we  find  the  values  of  the  functions  cos  is,  sin  ie,  in  terms  of 
cos  kg,  sin  hg,  and  the  <7  functions  just  given ;  always  noting  that  when  h  —  0,  we 
have  only  for  i  —  ±  1,  —  Je  as  the  value  of  the  function. 

We  can  employ  equation  (22)  when  only  a  few  functions  are  needed,  or  as  a 

check. 

A.  P.  s. — VOL.  xix.  G. 


50  A   NEW   METHOD   OF   DETERMINING 

It  may  be  of  value  to  have  y*  in  terms  of  zh  and  the  J  functions.     Prom  the  sec 
ond  of  equations  (20)  we  Jiave 

(0)  (1)  (2) 

z2    +  -J3A.*3    +etc. 


JK  .  z-1  -  -  i<72A  .  z~2  —  iJg,  .  z~*  —  etc. 


(0)  (1)  (2) 

-  Z"1  -  Z~2  •  *~3  +  etc. 


(2)  (3)  (4) 

—  J.z     —   J.*.£   —  cfe.       —etc. 


(1)  (0)  (1) 

2/+2  =       -  ye/A  .  Z   +  | «/2A  •  22   +  |^3A  -  &   +  etC. 

(8)  (4)  (5) 

-ty.z-1  -%J*.z-*  -KsA-^"8  -etc. 

(i)        (0)  _2     (i) 

(3)  (4)  (5) 


Then  from 


y~l  =  2  cos 


y*  —  y~l  =  2  V  —  1  •  sin  ie 

we  find  the  values  of  cos  e,  sin  8,  cos  2e,  sin  2e,  etc. 

In  case  of  the  sine,  as  for  example  when  i  =.  1,  we  have 


y  —  y~l  —  2  V  —  1  sin  e ;  but  in  z  —  z~l  =  2  V  —  1  sin  g, 


we  have  the  same  factor,  2  V — 1>  in  the  second  member  of  the  equation. 
From 

r  =.  a  (1  —  e  cos  e) 
we  find 

f- J  zr  1  —  2e  cos  e  +  e2  cos  2e 
^  V  =  1  +  2e  cos  e  +  3e?  cos  2f  +  ^e3  cos  3e  +  etc. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  51 

For   (- )   we  have 

\a/ 

(r  \ 2 
-  J  =  1  -f  2C"  —  2e  cos  £  +  Je2  cos  2e 
/ 

But 

d :_/rj\  _  2e  gin  e-  Q  _  e  cos  e  j  ^1  -  2e  sin  e, 
(Zgr  Vav  7  rfy 

and 

p      (0)  (2)-,  p      (1)  (3)-.  p      (2)  (4)-. 

sin  £  =  U7A   +  J,,       sin  ^  +  \  \  J^  +  J2A      sin  fy  +  i  I  Ju  +  ^A      sin  3^  +  etc. 

L  _l  L_  _l  l_  J 


Multiplying  by  2e  .  d^  we  have  for  the  integral  of  - 

(0)        (2)~i  9^>  r    (1)        (3)n 

+J*  J  cosg--^\_J2,  +  J2,  Jcos2#--3A  +    3A     cosS  —  etc. 


where  c  =  1  +  f  e2. 

By  means  of  (22)  this  becomes 


/r\2  (i)  (2)  (3) 

(a  )  —  1  -f  f  e~  —  f  eA  cos  g  —  ±  J2J,  cos  2g  —  ^  cos  3^  —  etc. 

(«»  \  —  2 
-J  ",  we  have 


Se2  .  cos  2e  ==  |e2  (1  +  cos  2e),  4e3  cos  3a  =  e3  (3  cos  e  -f  cos  3e), 

5e4  .  cos  %  =  |e4  (3  +  4  cos  2e  +  cos  4e),  6e5  .  cos  5e  z=  T%e  5  (10  cos  e  +  5  cos  3e  +  cos  5e), 

7e6  cos  6e  =z  ^e6  (10  +  15  cos  2e  +  6  cos  4*  +  etc.) 

and  hence 

~2  =  1  +  ¥  +  ¥*4  +  ^  +  etc. 
+  [2e  +  3e3  +  f|e5  +  etc.]  cos  e 
+  [f  ^  +  ^-e4  +  -y^e6  +  etc.]  cos  2e 
+  [e3  +  f|e5  +  etc.]  cos  3e 

Me°  +  etc-]  cos  4e 


52  A   NEW   METHOD  OF    DETERMINING 

Attributing  to  i  proper  values  in  equation  (29)  we  find  the  expressions  for  cos  e, 
cos  2?,  cos  3e,  etc.     We  then  multiply  these  expressions  by  their  appropriate  factors  and 

thus  have  the  value  of  (r\~' 

\aj 

(2) 


,r.  «  /r\~  c 

(a  j    ~  2—  &  COS  &  (a)      ~  2-«  Ei 


(2)  (-2) 

The  following  are  the  values  of  Rt  and  RL  to  terms  of  the  seventh  order  of  e. 

(2) 


J.TI  -  2e  -}-  -%e  -  -  - 

(2) 
-fl/  -  "  -&    ~~ 


/2 

(2) 


&  ~  ~  256  0  e 


(2) 

7?  —  _   .  _2  5  fP  _|_     625  .o? 

•^^S  —  192el4608e 

(2) 

-"6  —  -^O60 

(2) 

7?  —  _        2401   P1 

•**!  —  2"30406  ' 

^o"2=  =  1  +  e2  +  f  e4  +  -W  +  etc. 


(-2) 
r>       —    1 

J.4/4        - 


24  °  240^ 

(-2) 
J£       __    l  Q  9  7  ff> 1  6  6  2  1  g7 

(-2) 

r>      —   1223.06 

•**•     —      1  6  0  6 


See  HANSEN'S  Fundamenta  nova,  pp.  172, 173. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 

(2)  ( 2) 

We  add  also  the  differential  coefficients  of   R, ,    R,        relative  to  e. 

l    7  Is 

(2) 
dJRa  o . 


de 


53 


(2) 


de 


=  -  - 1*  +  tt*  —  ^Wo«6  ±  etc 


(2i 


w 

(2) 


4875^ 
e 


de 

etc. 

(-2) 

dR<) 

(-2) 

de 

(-2) 


re°  -4-  etc. 


etc. 

e  +  Se3  + 

2  +  f  e2  + 
5e  +  le3  4 


+ 
?4  + 


_ 
64 


(-2) 


de 

(-2) 


4608     ^ 


54  A   NEW   METHOD   OF    DETERMINING 

The  value  of  -,  found  by  integrating  d(r\  —  2e .  sin  e .  dg,  is 

(Jj  \Cu     / 


,ri  (1)  (2)  (3) 

-2  =  I  +  fc2  —  f  JK  cos  g  —  ft/a*  cos  %  —  tf  J"3A  cos  3g  —  etc. 


(2) 

In  terms  of  the  Rt  functions, 


r2  (2)  (2)  (2) 

-  ZT:  1  +  -le2  —  jRi  cos  0  —  ^o  cos  2q  —  R^  cos  og  —  etc. 

a2  2  *  y 


Again,  since 


Let 


then 


dg          r* 
we  have 


a2          T>(~2)         - 
2   =  ^-     cos  ^^ 
2 


and  hence 

^!~2)  = 


The  coefficients  represented  by  Ct  designate  the  coefficients  of  the  equation  of 
the  centre. 


THE   GENERAL   PERTURBATIONS   OF   THE   MINOR    PLANETS.  55 

Using  the  values  of  the  d  coefficients  given  by  LE  YERRIER  in  the  Annales  de 
I'Observatoire  Imperial  de  Paris,  Tome  Premier,  p.  203,  we  have 


f-g  =    4:  (I)  -  2  (|)3  +  |  (iy  +  JftL  (|)7  +  W  (|)o]  sin  g 
I)2  ~  V  (I)4  +  V  (l)c  +  If  (I)8  +  etc.         ]  sin  2g 


(I)4  -  W  (I)6  +*  ^iP  (I)8  -  etc.  ]  si 


(I)6  —  ^W^  (I)8  +  etc.  ]  sin  Qg 


_l_   [10661993  /_«  \9  "1      •      Q 

|_10080V2/  J   sm  yy 

Converting  the  coefficients  into  seconds  of  arc,  and  writing  the  logarithms  of  the 
numbers,  we  have  for  the  equation  of  the  centre, 


+  [5.9164851  (I)  —  5.6154551  (|)3  +  5.5362739  (|)5  +  5.787506(|)7  +  6.  25067  (|)9]  sin  g 

+  [6.0133951  (|)2  —  6.1797266  (f)4  +  6.067753  (|)6  +  5.59571  (|)8]sin2^ 

+  [6.2522772  (|)3  —  6.6468636  (|)5  +  6.690089  (|)7  —  6.22336  (|)9]  sin  3g 

+  [6  5491111  (|)4  —  7.093540  (|)6  +  7.27643  (|)8]  sin  4# 

+  [6.8775105  (|)5  —  7.533150  (|)7  +  7.82927  (|)9]  sin  5g 

+  [7.225760  (|)6  —  7.96973 

+  [7.587638  (|)7  —  8.40484 

+  [7.95944  (|)8]  sin  8# 

+  [8.33880  (|)9]  sin  9g 


56 


CHAPTER  III. 

Development  of  the  Perturbing  Function  and  the  Disturbing  Forces. 

By  means  of  the  formulas  given  in  the  preceding  chapter,  the  functions  ^•(^), 

ft  .  a2(;!)3j  etc.,  can  be  put  in  the  desired  form.    The  next  step  is  to  determine  the  com- 

plete expression  for  the  perturbing  function,  and  also  the  expressions  for  the  disturb- 
ing forces. 

If  k2  is  taken  as  the  measure  of  the  mass  of  the  Sun,  and  m  the  relation  between 
the  mass  of  the  Sun  and  that  of  a  planet,  the  mass  of  the  planet  is  represented 
by  mk2. 

If  x,  y,  z,  be  the  rectangular  coordinate  of  a  body,  those  of  the  disturbing  body 
being  expressed  by  the  same  letters  with  accents,  the  perturbing  function  is  given  in 
the  form 


i        xx'+yy'-\-zzf 


~l 
J 


"  1+m     J  r'3 

Now 

A2  =  (af  —  x)*  +  (y'  -  1/)2  +  O'-z)2, 


hence 

•  n  =*[*_'* 

1-f-m  L^J        r'z 
If  a  fl  is  regarded  as  expressed  in  seconds  of  arc,  and  if  we  put 

s  =  206264".  8,    p  =  -=£-.«,    «=-, 

1-J-m  a  " 

we  have 


THE   GENERAL    PERTURBATIONS   OF    THE   MINOR    PLANETS. 

Finding  the  expression  for  (H]  first  by  the  method  of  HANSEN,  we  let 
h  —  !'u,  .  k  .  cos  (n  —  K),        h'  =  **  .cos<p.  cos  ^  .  ^  .  cos  (n  — 


I  —  '*  .cos<p.Jc.  sin  (n  —  K  ),  I  —  !\_  .  cos  $'  .  ^  .  sin  (n  —  K}), 
and  have,  if  we  make  use  of  the  eccentric  anomaly, 

(77)  =  h  .  cos  e^)2  .  cos/'  —  ehffi  .  cos/'  —  I.  sin  e  .  (£)2  .  cos/' 

,     7/  /a'\2     sin/"'  7,/a'\2    sin/'         7,       .         /a'\2     sin/' 

-  I  .cos  e(   J   .        -  ,  —  el'(   ,     .          ,  -\-hf.  sm  e  (   ,)   .     —  ^ 

\r/       cos  ?>  \r/      cos  ^  \r  /        cos  ^ 


Putting 


cos/'  =  y\  .  cos  g'  +  y'2  .  cos  2g'  +  y'3  .  cos  3g'  -f  etc. 


=  ^  .  sin  </'  +  3'2  .  sin  2gr'  +  «V  .  sin  3</  +  etc. 


we  find 


cos 


'2(hy'.2  —  A'5'2)  cos 


2(A/2  4 
etc. 


'x  cos  (  —  </       )  + 


e) 


'*    cos 


''i  —  I'b'i)  sin  ( — g'  —  e) 
eU\  sin  (—  g'        ) 

2 —  Z'^'.J  sin  ( — 2g'  —  F) 
sin  ( — 2^r'       ) 
Z'^'2)  sin  (     (2g'  —  e ) 
etc., 


where 


(0)  (2) 


(0)  (2) 


']- 


etc. 


etc. 


57 


A.  P.  s. — VOL.  xix.  H. 


58  A    NEW   METHOD    OP    DETERMINING 

When  the  numerical  value  of  (H)  has  been  found  from  this  equation  we  trans- 
form it  into  another  in  which  both  the  angles  involved  are  mean  anomalies.  For  this 
purpose  we  compute  the  values  of  the  J  functions  depending  on  the  eccentricity,  e,  of 
the  disturbed  body  just  as  has  been  done  for  the  disturbing  body.  The  values  of  the 

(0)  (1) 

J  functions  can  be  checked  by  means  of  the  values  of  JJ,A,  <//iA,  given  in  ENG  EL- 
MAN'S  edition  of  the  AWiandlungen  von  Friedrich  Wilhelm  Besxel,  Erster  Band,  seite 
103-109,  or  by  equations  (30)2. 

Thus  by  means  of  the  equation 

(m+l)  (m-l)_        m  (.,„) 

VIA       ~r  ^/>\       —  T-T  •  «A 

fl.A 

(m)  (0)        (1) 

we  are  enabled  to  find  JhK  if  JhK,  JhK  are  known. 

It  must  be  noted  that  the  argument  of  BESSEL'S  table  is  2 .  h%,  or  2 .  fcl,  or  he. 

a) 
Thus  if  it  is  sought  to  find  the  value  of  J2X ,  we  enter  the  table  with  2 .  2A,  or  2e  as  the 

argument. 

When  we  need  the  functions  for  k  from  h  =.  —  1  to  It  —  4,  we  must  find  the 

(3)  (2)  (1)  (0)  e  (-2) 

values  of  $J  ' ,  ±J      \J    ,  \Je  ,  —  -,  and  -  -  \J  e. 

42<i2-i22  *  2" 

(1)  (0)  (3) 

The  values  of  ^  .  J    €  and  Je    we  take  from  the  table.     To  find  J  c  we  have 

2  '  2"  2"  4  2" 

(3)  (1)  2  (2) 

**     e    —  ^    «    ~^~    ~     c"  •  v 


JD  2     I-  (0)  1  (I)-, 

—    — v     e    "T     r~Tl  e/    e    +   ~T  •  J .  e 

4-%  4.TL  4  T  4¥         4^J 


For  «/  e  we  have 


(2)  (0)  JL_  (1) 

o«    — J ~e    ~T    o    e-t/C. 

- 


(2) 

And  for  Je  we  have 


(2)  (0)  1          (1) 

e_     —  Je^    +  ~<rJe_ 

2  2  2         2 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  59 

The  expression  for  (H)  can  be  put  in  a  form  in  which  both  the  angles  are  mean 
anomalies.     Thus,  resuming  the  expression  for  (H), 


sin/7 

COS  CP' 


(H)  =  h  .  cos  e  f  a,J    cos/'  —  e/i  ( a,  J    cos  /'  —  I .  sin  e  f a,  J   .  cos  /' 

+ 1  .cos ,  (°Y. sin/;  -rf'(a;)2.  ?!£/;  +  A-.8in  E.  CY  . 

VrV       cos  /  \r'/       cos  <p'  \r' J 

in  which 

A  =  ~.^.cofl  (n — JfT) 

a2  v 

A'  ZZ    ^ .  COS  <£  .  COS  ^)'  .  &!  .  COS   (II  -  -  K^)      ZZ      Jff  .  - 
7  At  7  /r-r  TT"  \  1  V    Sin 

t  —  - , .  cos  <h  .  A; .  sin  ( II  —  7t  )    zz    sw .  - 

a"  a 

)  .  A^j 


we  find  the  expressions  for  (~,Ycosf9  f^J         -^,  as  follows.     "We  put  as  before 
(p)  "  cos/  =  /!  cos  gr'  +  7'2  cos  2#'  +  y's  cos  3g'  +  etc. 

(r'T  cos  V    =  5/1  Shl  ^  +  ^  8ln  ^'  +  ^  Sln  ^'  +  etC" 


r 

If  we  differentiate    ,  cos  f  relative  to  q'  we  have 

a  J  y 


_  cos/'     <tf  __  r'     gin  f    df  _  __  sin 
a'     '  dg'        a!  '         •'   '  dg'  ~  cos 


dr'         aVsin/'          df         a'2 

since  —  7  —  -      —  f-,         /    zz  -.,  .  cos  of)  ; 

d  cos  <'  d'         rn 


and  hence 


0 
—  —  ^  .cos/. 

'2  ^ 


60  A   NEW   METHOD    OP   DETERMINING 


Similarly,  in  the  case  of  -         _    we  have 

J  ?  a'  cos  <p' 


_#  /V  sin  f\   _  _  _  of*      sin  ./" 
dg'2  \af  cos  <p'J   "  r'2  '  cos  ?>'' 


T>    *  r  ^  j  r   sn  •       / 

out   -  cos  /  rr  cos  e  —  e,     and  -.  —  sin  r 

a'  a'  cos  <p' 


Hence 

_^2Ccos/')  _  a^ 

"  r'2      >S  * 

a'2      sin        _ 


cos 


(»)  (2)-|  r       (1)  (3)  -I 

cos  e'  =  —  ^  +  —  «/A,  J  cos  <;'  +  J  L«S^  —  «4,  J  cos  2gr'  +  etc. 


r    (o)         (2)~|  r    (i)          (3)-i 

sm  f'  \jfv  +  JK,  J  sin^'  +  j[J,A,  +  ^  J  sin  "2gf  +  etc. 

From  the  values  of  cos  e'  and  sin  e'  we  have 

a'2  r  T(Q}    I2n  ,  r  (1)      (3)~i  r  (<2)      (4n 

,T,  COS  /       :    |_e7v  —  J,,  J  COS  #'  +  2    |y2A  --  e^  J  COS  20'  +  3  [^  --  e/^  J  COS  3(/'  +  etc 

a'2  sin  /•'        r   (0)      (2)~i  r   w       (:5n  r   (2> 

rCsi  =  L^'  +^'J  sin  ^  +  2  L^'+  ^  J  sin%'  +  3  L^'+ 

We  now  assume 

* 

(i~i)     (ui)i 
A   +  ^ 


.>*_    f^*  —  ^* 


(i'-i)          (t'+i)-i  p    (i'-n          (i'+i)-i 

-    -^   J     «^=r[^-.+  ^]. 


Comparing  these  expressions  for  y'?,   S'i>9   with  those  found  in  the  expression   for 


'2  *          -/*' 

^2  •  S~    —  /  given  above,  we  see  that  the  relation  between  them  is  i'2. 

r2     cos  (    * 


9° 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  61 

The  expressions  for  cos  F,  sin  F,  are  the  same  as  those  of  cos  F',  sin  F',  if  we  omit 
the  accents. 

Hence  if  we  perform  the  operations  indicated  in  the  expression  for  (H),  we  have 


-  i- 
cos'  (±  ig  —  »V)  —  ^  WtfvS&r#*'}  sin  (±  *7-*V)     (2) 


y*  and  i'  having  all  positive  values. 

Attributing  to  i  and  i'  particular  values,  we  find,  noting  that  $0  =•  0,  and  8'0  =  0', 

-^  \]i  .  w\  +  Wx  ]  cos  (  g  -  -  g')  -  -  \  [Ul7\  +  Z'y^'J  sin  (  g  -  -  g>) 
1P-71/1  —  *'W'i  ]cos(—  flr--  3r')-j-4[>Vi  —  /VA]  sin  (—  flf--  5r') 
P-7o-/i  cos(  —  ^')  -  -  i  Z'^'i  sin(  —  ^') 

2  [A  .  y,/2  +  A'.  W  J  cos  (  5r  —  2gr')  -  -  2  PV*+  Z>^2]  sin  (  ^  —  2g') 
2  [A.yi/2  —  A'.  WJ  cos(—  g  —  2g')  ±  2  [Z.V2—  ^'J  sin  (—  </  —  2<?') 
2/i.W'2  cos(  —  2^)--2Z'.^'2  sin(  —  %r') 

I  [*  •  yi/3  +  *'.  ^^'3]  cos  (  ^r  -  3^/')  -  -  f  P^^+Z'.y^]  sin  (  g-  3g') 
etc.  —  etc. 


'.W'J  cos  (—  20  —  0')  —  i[Z.Vi—  f-y^'J  sin  (-2^  —  ^ 


The  numerical  value  of  (If)  given  by  (1)  must  first  be  transformed  into  a  series 
in  which  both  the  angles  involved  are  mean  anomalies  before  it  can  be  compared  with 
the  value  given  by  the  equation  just  found. 

If  we  find  the  value  of  (Z7)  from  the  preceding  equation,  it  can  be  checked  by 
means  of  the  tables  in  BESSEL'S  Werke. 

The  expression  for  {.i  fa.}  is  known;  and  with  the  expression  for  (If)  just  given, 
we  obtain  the  value  of 


The  next  step  is  to  obtain  expressions  for  the  disturbing  forces. 


62  A   NEW   METHOD    OF    DETERMINING 

Let  v  the  angle  between  the  positive  axis  of  IK  and  the  radius-  vector  measured  in 
the  plane  of  the  disturbed  body,  here  called  the  plane  of  X.  Y.     The  differential  coeffi- 
cient of  the  perturbing  function  H  relative  to  the  ordinate  Z  perpendicular  to  this 
plane  is  found  by  differentiating  LI  relative  to  z  and  afterwards  'putting  z  =  0. 
Thus  from 

O   _      ™'      P  .  _  rr'     „-] 
'  l"+m  LA         r*' 

_     m'      Fl         xx'-\-  yy'~\-  zz'~\ 
~  ~~^  J' 


A*  =  (x-i*)*  +  (y-y>?  + 
=  r*  +  r'2--2rr'  H, 

we  find 

dQ   _      m'      I"        _!_      dJ  _   _  '  r        dH~} 

dv    '  '   l-\-m  L        A*  '  dv          r"       dv  J 

dQ  m'      T        1    (r  —  r'm      _  H~\ 

dr^  ''  :  r+^i  L    "  T2  V        J       j  "      72J' 

j/-v  w*'      r         1     j  A  >  dz~l 

d£l  —  -  --  To-^A  —  z'.  /3  L 

l-fm  L        ^  r3J7 

—  _ 

A  dA  ,dH  A  dJ  ,Tr  dJ  2' 

A  —  rr  —  rr        ,  A       zr  r  —  r  //.  -  rz  -- 

dv  dv  dr  dz  A 


Hence 


dtl  _  m' 

dv   ~  1-fri 


dti  _ 


dr    '  l-f 


'     f  1  1  ~1       ,  r -T  ni'         r2 
—             ff   H  —  _ 

m  L  A3         r'*_\  1-fm  '    A3 


<Zi2  7H'      Fl  1    ~1       •         T        i      ' 

,„  =  —  T  ,        -?5  —  —3    sin  *•  r  s 

d^  l-fm  L  A3        r/JJ 

where 

H'  -  sin  (/  +  n)  cos  (/'  +  H')  —  cos  /cos  (/+  II)  sin  (/'  -f  IT) 
2'  =  —  r'  .  sin  Jsin  (/'  -f  "HO- 


THE   GENERAL   PERTURBATIONS   OF   THE   MINOR   PLANETS.  63 

As  before  the  origin  of  angles  here  is  at  the  ascending  node  of  the  plane  of  the  dis- 
turbed body  on  the  plane  of  the  disturbing  body,  and  the  plane  of  reference  is  that 
of  the  disturbed  body. 

If  we  differentiate  the  expressions  for  r    ",  (,  ',  we  find 


dr 


d,2&  dQ  m'          3 

i  2   +  T  ~i  -—  ,      -  V-n= 

dr*  dr        1  +m         J5 


+    »'   (-I—A-,)  rr'H-2^--.^ 

l+m  \/J3          r3/  l+m      /J<i 


l+m  Vd3          r'3J  i+1 

l+m  ' 


r  —^  =  ,^~  .  4  (r2  —  rr'H)  sin  Jr'  sin  (/'  +  IT) 
-  6  v  w  / 


m'  3         .        o  7-     ,o  .;>/    /.,      .     T-T.X 

=  r     -.  —  ?  sm  ~Ir'z  sin  -(  /'  +  II') 
l?/i       J5  v>/ 


i  =  rr-  (  Va  —  A-,)  sin  1.  r  sin  (/+  n) 

l+m  V  J3          rv  Vt/ 


sm    -r  sm 


m'        3        .     97-        ,    •     /  a'  i    TT\    •     /  ^-/    i    TT/\  m'    /  1          1 

•      •  sm  "J-  rr  sm  (  f  +  n)  sm  (/  +  n  }  +  —      cos 


To  eliminate  ^T  from  some  of  these  expressions  we  find  from 

that 

_    i*+r»          j^ 
2J»         "  2l 


The  expression  for  r —  then  becomes 

dr 


->  y-^  /  iT~~            f*>.                    9                                 T                                                                   ^^ 

di2  m  i    r 2 — r2           1             r    TT\ 

" —    — -.--  ti 

dr  14-m  I       24s             2J          r'2      J 


dr 

From  the  value  of  A2  we  have,  further, 

r^—rr'H  r'2— r2 


2J5 


64  A   NEW    METHOD    OF    DETERMINING 

and  hence 


*-£  _   in  Bin  j  ^  Bin 

J5  J:1  J 


_ 

2 


the  latter  of  which,  by  means  of  the  expression  for       ,  becomes 


'    |~  r'2  —  r2  1   ~|  r  /  /.   .    T^N  m'        .      r  r      .      /   ~   , 

--  sin  Jr  sin  (f+  H)  —  sin  I  --  sm  (f—  II) 

m  L      J5  3J:!J  1m  r'3 


The  expression  for  A2  also  gives 


(r^—rr'HJ  _    (r'2—  r2)2         _^^_     i     J 

» 


4J6 


by  means  of  which  we  find 


-         m'    l"3(r/2—  r2)2  _     /2     i     J_~|_     ^_     J^      rr 
'~  Im"  L      4J^~         "   J3  "       4  J  J        1+m  '  r72  ' 


. 
Tdr'~     + 


If  we  put,  for  brevity, 

(J)    =    ,.  sin  J°;2  sin 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  65 

the  expressions  which  have  been  given  for  the  forces,  together  with  the  perturbing 
function,  are 


«r"  d*S 


+  «»•('!")  =   ^(".Y 

\<tr  /  \  J/     L  a  «    a, 

\  drdZ/  \  A)     L  a'2         a2  a2J        a        a' 


a\r>    sin2/     r'2       .     0/   /.,    .    nA  /o\ 

J  •  -it  •  a-  •  sin  -u  +  n  )  -  K  J 


•> 

a 


f  9  sin       r     •      /  *    , 

aa       =       •  -  "    sin      n  - 


•  • 

drdZ'  \J/       a'2         a2    a2          a        a 

«\3     sin  / 


•  r  sin  (/+  n) 

Vt/ 


i2/«\3     sin  /  r     .      / 
^  U    •  ~^~  a  Sm 

sin2/     r'    •     /  /•,    ,     r-r.x  r     .     /  /»   ,  o/a\3cos  / 

' 


The  form  given  to  these  expressions  is  the  one  best  adapted  to  numerical  compu- 
tations ;  and  the  equations  are  readily  derived  from  the  preceding  in  which  the  magni- 
tudes occur  in  linear  form.  , 

Thus  from 


r          __  __  _       1  _r     TJ  "1 

~  "  2J  "  "   r'* 


'9  9 

*  - .  ^  y** 

rfr       "  1+m   L     2J»~ 
A.  P.  S. — VOL.  XIX.  I. 


66  A   NEW   METHOD    OF    DETERMINING 

we  have 


fji     ra3     r'2  __  a3       rH         /z     a          j*_    ^a"       2     r       rr 
Z          2    La**    J*         a''~JsJ~     2*J      "a"  *r"*(     'a* 


where,  as  before, 

/^      /«'\2    r      TT  a' 

rz  -  .  (  -t  )   .     .  H.  a=- 

a      \rj      a  a 


. 

1-fm 

In  a  similar  manner  all  the  other  expressions  for  the  forces  have  been  derived. 
When  we  compute  only  perturbations  of  the  first  order  with  respect  to  the  mass 
we  need  the  perturbing  function 


=  p  C 

and  the  forces 


1     »J~1         I    fa\ 

"  ;,-a,]  "  If  (  J  -  - 
£sin  (/  +  n')  +  (/) 


The  other  forces  are  only  needed  when  we  take  into  the  account  terms  of  the  sec 
ond  order  also  with  respect  to  the  mass. 

An  inspection  of  the  expressions  for  the  forces  shows  that  besides  the  functions 


we  need  expressions  for  the  magnitudes 

'\2         1     r2       sin  /  rf     .      /  /.»        T-r,\       sin  I  r 


(r'\  1     r2       sin  /  rf 

a')    '     *'*>-—* 


a 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  67 

When  these  are  known  we  multiply  the  function  ^a2(^j    by 

F/r'V2         1       r2~|  sin/     r'    •     /  />    ,    T-T,\  sin/r    .      /  /•    ,    T-T\ 

( •/)  -     2  •  -i  U  •  -/sinff  +  n'),  sm  (/  +  n), 

L  Va  /  a       a2J  a         a  a    a 


the  function  fj.a  (a  J    by 


rn  i    r2-]2  Q    cin  r  r'  .    rV'2  1     ™s 

F-?3-  I™  >in(/+n/)&-i:= 

3  ^  .  ^ .  sirf  (/  +  no,       |  *=-'  r  sin  (/  +  n)  g  -  i,  H 

t*  ^  €*          t*  L    t/>  6t     Ci      J 

3  ^  r  sin  (/  +  n)  ^sin  (/  +  H'). 

«i  /T.  \«/  /  fj'  \<J  / 


We  will  now  find  the  expressions  for  (/),  (/)',  (^)",  and  for  the  various  factors 
just  given,  that  are  the  most  convenient  for  numerical  computation. 
We  have 


(J)  =  ?  sin  /.  Bin  (/  +  !!'). 

Putting,  for  brevity, 


—  -„  cos  d>'  sin  /  cos  II' 

a2 

I'  -          l~  sin  I  sin  IT, 


and  noting  that 

/«'\2<,iTi  /7      r   <°>       (<2n  r  (l)       (3)~i 

(1)   ^£  =  [^  +  Jv  J  sin  flf  +  2[y2V  +  ^  J  sin  2g'  +  etc. 

/a'\2  r  (o)      (2)~i  r  (1)      (3)~i 

(-,)  cos  f  =  [Jv  —  Jx  J  cos^'  +  2|_/2V  —  J^  J  cos  2g'  +  etc. 


68 

we  have 


A   NEW   METHOD    OF    DETERMINING 


r       (0)  (2)  -I  [-      (0)  ('2)-j 

(1)  =  I    |yA,  +  ,/A,  J  sin  (-  -   g')  +    I'  [y*  —  JA,  J  cos  (-     0') 

r  o)       (3)~i  r  (l)       (3n 

+  26  [_<72A,  +  ,72A,  J  sin  (—  20')  +  26'  |yw  —  '^'  J  cos  (—  2^') 

r      (2)  (4)~1  [-       (2)  (4)~1 

+  36  [_JW  +  e/3V  J  sin  (—  3g')  +  36'  [_  J3A  —  ^  J  cos  (—  80') 
+  etc.  +  etc. 

The  value  of  (/)'  is  found  from 


From 


r' 


-  1  — 


cos  e 


we  find 


Expanding, 


+  (fe'2+|e'4    +  etc.)  cos  2^' 

+  ^e'3  cos  30'  +  ^J-e'4  cos  40'  +  etc.  ; 

which,  for  brevity,  we  write, 

(~>)°  =  PO  +  2  pi  cos  0'  +  2  p2  cos  20'  +  2  p3  cos  30'  +  etc. 
But 

r      sin  f  (0)  (2) 


(2)~1  f     (1)  (3)~1 

+  ^  J  sin  ^  +  i  L^A  +  J<*  J  sin  %  +  etc. 

r  f     (0)  (2)-i  r     (i)  (3)-i 

~  .cos/  =  —  fe+  [_j;  —  Jx  J  cos0  +  I  |y2A  —  J^  J  cos20  +  etc. 


(3) 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


69 


Putting 


I   =  ~ f  .  cos  <p  sin  /  cos  IT, 


r   w        (3 

'.,  —   I     ,/2A  e/£ 

etc. 


w  = 


sin  </ 


etc. 


For  (/)"  we  have  the  expression 


Putting 


we  have 


Zi  -  :  as  •  sin  /  sin  IT, 

2  2    |_f    2A    T-   '    2A   J 

etc., 

-f^e.po  ^ 

I         7 

+  ?i .  p!  ^    COS  ( —  g  —     g' 

-  2  hep!      cos  (  g' 

H~  ^i  •  p2  •  y\  cos  (     ^  —  2g' 

~\~  ^l  •  Ps  •  7i  COS  ( —  <7  —  2of' 

\  »/  «y 

-22^.  p2  cos  (          -  2g' 
±  etc. 


—  2 .  -£  cos  J,     and  using  the  pf  coefficients  as  for  (/)', 


"  z  :  **  '^  +  ia  .  P!  cos  (—  g')  + 


(4) 


To  obtain  an  expression  for  the  factor 
have  that  for   f-V. 

\a/ 


.  p2  cos  (—  %r')  +  etc.  (5) 

^>)2-      *2  -H  it  is  only  necessary  to 


70  A    NEW   METHOD    OF    DETERMINING 

In  terms  of  the  eccentric  anomaly  we  have,  at  once, 


:  :  1  —  2e  cos  6      er  cos  2 

a 


=  1  +  |e2  —  2e  cos  e  +  Je2  cos 
Substituting  the  values  of  cos  e,  and  cos  2e,  we  have 

r\2  (1)  (2) 

-;  =  1  4-  fe2  —  f7*   cos  0  —  |  J"2A  cos  2#  — 
To  find  an  expression  for  the  factor  ^L  .  ^  sin  (/  +  H'),  for  brevity,  we  let 

O.  CL 

,  _  sin7 


(1)  (2)  (3) 

"  —  etc. 


_        _       cog     , 


?*    ^i  n    /  i  /4 

and  from  the  known  expressions  for    -,  -—A,     ->  cos/  ,    we  get 


a        a 


r    (o)        (2)~i  r    w        (3)~l 

I  J«  +  J*    \  *  sin^r'  +  J  [^  +  J,v  J  Cl  sin  2gr'  +  etc. 

•—  —  ' 


r    (0)         (2)-i  r    (i)          (3)n 

—  |e'c2  +  L^A'  —  J*  J  G,  cos  g'  +  i  L^ix'  —  J-*  J  C2  cos  2^r'  +  etc. 

In  the  same  way,  if 

sin  I  sin  /       .     ^ 

ft,  =  -        .  cos  ct>  cos  n,  c4  =  -     -  .  sin  IL 

a  a 

we  find 


(0)         (2)-i  r    (i)         (3) 


(6) 


By  means  of  the  expressions  for  the  factors 
(.-)'• 


THE   GENERAL   PERTURBATIONS    OF   THE  MINOR  PLANETS.  71 

just  given,  we  can  form  those  for 

3       -'2         1    r2~2 


3  p-'2  _     1    r2~| 

4  La72         a2  a*J 


3    sin 
2    ~~~a 


3    sin2  7       r'2    .    2  /  /,    , 
-r-  .  -.sm-  (/>  +  n' 


+  n) .    sin  (f  + 


72  A    NEW    METHOD    OF   DETERMINING 


CHAPTER  IY. 

Derivation  of  the  Equations  for  Determining  the  Perturbations  of  the  Mean  Anomaly, 

the  Radius  Vector,  and  the  Latitude,  together  with  Equations  for  Finding 

the  Values  of  the  Arbitrary  Constants  of  Integration. 

HANSEN'S  expressions  for  the  general  perturbations  are 


cos* 


where 

/—  o)  — 1  +  2-     h*p      -  [cos  (/  —  «)- 


In  this  chapter  we  will  show  how  these  expressions  are  derived  from  the  equations 
of  motion,  and  from  quantities  already  known. 

The  equations  for  the  undisturbed  motion  of  m  around  the  Sun  are 


dt* 


(1  +  m}  y~  -  0 


-?    +  V  (1  +  m)  *-•  =  0 

dt*  }  r* 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  73 

The  effect  of  the  disturbing  action  of  a  body  m'  on  the  motion  of  m  around  the 
Sun  is  given  by  the  expressions 

,79/V  -  X  x'\  ,79/V  -  V  y'  \  ,T>(  '  Z'  -  Z  Z'  \ 

mKr(  -  ;),  wifcv     y      -"7-),  m,'Tc(  ,). 

\    A3  r«/J  \    J3  r'3/'  V    J3  r'3J 


Introducing  these  into  the  equations  given  above  we  have  in  the  case  of  dis 
turbed  motion 


dt* 

(1) 
dP 


dt2  '    r3 

The  second  members  of  equations  (1)  show  the  difference  between  the  action  of 
the  body  m'  on  m  and  on  the  Sun.  The  action  of  any  member  of  bodies  m',  m" ',  m"', 
etc.,  can  be  included  in  the  second  members  of  these  equations,  since  the  action  of  all 
will  be  similar  to  that  of  m'. 

The  second  members  can  be  put  in  more  convenient  form  if  we  make  use  of  the 
function 


m'    /I          xa/-\-yy'-\-zz'  \ 
+m 


Differentiating  relative  to  x 

dtt  _      m!    (_      1     d^ 

Jx  -  -  i+m  \         j"  '  d.x 

But  since 

A2  =  (x'  -  xy  +  (y  - 

we  have 


<LA 
dx 


A.  P.  S.  —  VOL.  XIX.  J. 


74 

and  hence 


A   NEW    METHOD   OF    DETERMINING 


a\  dtt             ,  (x' — x          x'  \ 
+  m)  —  =  ra'  f— ). 
/  dx                \     A3              r'V 


In  the  same  way  we  derive  the  partial  differential  coefficients  with  respect  to 
y  and  z. 

The  equations  (1)  then  become 


+  Jc2  (1  +  m)  -3  ==  tf  (1  +  m) 


dy 

dQ 
dz 


(2) 


Let  X,  Y,  Z,  be  the  disturbing  forces  represented  by  the  second  members  of 
equations  (2), 

It,,  the  disturbing  force  in  the  direction  of  the  disturbed  radius-vector, 

S9  the  disturbing  force,  in  the  plane  of  the  orbit,  perpendicular  to  the  disturbed 
radius-vector,  and  positive  in  the  direction  of  the  motion. 

If  f  be  the  angle  between  the  line  of  apsides  and  theradius-  vector,  the  angle  be- 
tween this  line  and  the  direction  of  8  will  be  90°  +  /./We  then  have 


In  case  of  J?,  we  have 


and  for  S, 


From  these  we  find 


r  r  ' 


y   i    o  x 

-   -r-   O  —  . 
r  r 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  75 

If  we  wish  to  use  polar  coordinates  we  have 

da 


dx 


=r  R  cos  f  —  8  sin  / 


—  -  =  It,  sin/  +  JS  cos  /. 

From 

x  —  r  cos  /,       y  =  r  sin/", 

we  find 

dx  =  dr  cos  /  —  rdf  sin  / 

dy  zz  dr  sin  /  +  rdf  cos/' 

d2x  =  d2r  cos/  —  rcPfswf  —  2dr  dfsm  f  —  rdf  2  cos  / 

d?y  =  d2r  sin/  +  rd2fcosf-\-  2drdfcoaf  —  rdf2  sin/ 

From  the  expressions  for  dx  and  dy  we  find 

dy  cos  f.  —  dx  sin/=  r  df 


dxcosf.  -f  (Z?/sm/— 
and  hence 


dtt  I     dQ    .     »  ,    dQ          j. 

— -  = .-r^sm/-    -^  cos/ 

rfa;  r      ^/  rfr 

d&  I     dQ  £  .     dti     .      r 

-  —       -  -  -j*  cos  /  +  -  -  sin/: 

(Zy  r     df  dr 


from  which  we  see  that 


If  we  multiply  the  expression  for  d?x  by  cos  /,  that  of  d2y  by  sin  /  and  add, 
we  obtain 


d?x  cos  /  -f-  ^22/  sin  /  —  d2r  —  r  df2. 


76  A   NEW    METHOD    OF    DETERMINING 

In  a  similar  manner  we  find 

d2y  cos  f  -  -  d2x  sin  /'  =.  r  d2f  -\-  2dr  df. 

Operating  on  equations  (2)  in  the  same  way,  we  have 

d*x  /•    ,     d'2it      •       /.    ,     &2(l-j-m)  -*-r  /          -T7-     •       /•         73 

-55.  cos/  +  -=|  .  sm/  +  '  —  X.  cos  /  +  Y.  sm/  =  R 

dt*  ''  dt  r1  J 

-       .sin/  =  T.cosf-X  duf=S 


Comparing  the  two  sets  of  equations,  we  have 

2*f          =f(l  +  i»)i" 

dt      dt  ;  r  df 


(3) 


The  second  members  of  equations  (1)  and  (2)  are  small,  and  in  a  first  approxi- 
mation to  the  motion  of  m  relative  to  the  Sun,  we  can  neglect  them.  The  integration 
of  equations  (2)  introduces  six  arbitrary  constants ;  and  the  integration  of  equations 
(3)  introduces  four.  These  constants  are  the  elements  which  determine  the  undis- 
turbed motion  of  m  around  the  Sun.  Having  these  elements,  let 

«0  the  semi-major  axis, 

n0  the  mean  motion, 

g0  the  mean  anomaly  for  the  instant  t  =.  0, 

e0   the  eccentricity, 

<2>0  the  angle  of  eccentricity, 

7t0  the  angle  between  the  axis  of  x  and  the% perihelion, 

v0  the  angle  between  the  axis  of  x  and  the  radius- vector, 

/o  the  true  anomaly, 

e0    the  eccentric  anomaly. 

These  elements  are  constants,  and  give  the  position  of  the  body  for  the  epoch,  or 
for  t  =  0.  Let  us  now  take  a  system  of  variable  elements,  functions  of  the  time,  and 
let  them  be  designated  as  before,  omitting  the  subscript  zero,  and  writing  #  in  place 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  77 

of  7t0.      The  former  system  may  be  regarded  as  the  particular  values  which  these 
elements  have  at  the  instant  t  =  0. 

In  Elliptic-  motion  we  have 


—  e  —  e  sn  e 
r  cos  /'  =  a  cos  e  —  ae 
r  sin  f  =.  a  cos  <£>  sin  e 

v  -  f  +  X 
a?n2  -  F  (1  -f-  m) 

ISTow  let  n&  be  the  mean  anomaly  which  by  means  of  the  constant  elements  gives 
the  same  value  for  the  true  longitude  that  is  given  by  the  system  of  variable  elements. 
Further,  let  the  quantities  depending  on  n^z  be  designated  by  a  superposed  dash,  and 
let  the  true  disturbed  value  of  r  be  given  by  the  relation  r  =  r  (1  +  v). 

We  have  then 


~  e  —  e0  sm  i 
r  cos  f  rr  a0  cos  e  - 
r  sin  f  =  «0  cos  <£>0  sin  e 

V   =  f  +  7t0 

a      77  ~~     #*"'   I     I        1 171    1 

Q    /C»Q       .^—     A/      I   JL       j        II  v  I  • 

We  will  now  first  give  BRLTNNOW'S  method  of  finding  expressions  for  the  pertur- 
bation of  the  time,  and  of  the  radius  vector. 

Neglecting  the  mass  ra,  multiplying  the  first  of  equations  (1)  by  y,  the  second 
by  x,  we  have 

dy  dx 

C  being  the  constant  of  integration. 
Introducing 

cos  /'  —  x-*  and  sin  f  '=.  -. 

/  /i»7  *  r' 


78  A   NEW   METHOD    OF    DETERMINING 

into  equations  (2),  neglecting  the  mass  w,  we  find 

d*x  _|_  k*.  cos  f  -^~ 

dt*  7*2 

(4) 

(i  If       t       L '  .  Sin  f      -rr 

j  ,n  \  n  ^^~" 


We  have  also 


dx  v«   dr  /.    df 

—   /~>nQ   T  ^—  i*  si  n    r     _±_ 

i,       -^—     l_'*JO    /     .  ~~z~.  /     Olll    /     •     -    -' 

dt  •*     dt  •*.     dt 

dy  •       /•    dr  /.    df 

jr  =  *mf-jr  +  rc™f-^r 


dt  J  'dt  "^  'dt 

and  hence 


or 


and 


In  the  undisturbed  motion  we  have 


being  the  semi-parameter. 
Hence 


THE   GE^EKAL   PERTURBATIONS   OF   THE   MESTOK  PLACETS.  79 

From  these  relations  we  derive 


(5) 


and  also 

'  " 


i/P»  —  I  _      -1—  f  V'fr  .Qr   dt 

^=r    -    X  —     I     —  =^.   A}/    .  U»l>  //»\ 


If  we  eliminate  —  from  equations  (4),  noting  that 

1-^  =  l-- 

p         dt  k    p 


we  have 

dz          fcsin/ 


p 


neglecting  the  constants  of  integration. 
Since  r  rz  r  (1  +  v),  we  have  also 


=  x 


The  equations  (7)  then  become 


. 
y      dt  dt  ^/p~        J  \  p 

From  the  equations 


=  a0  cos  F  —  OQ^O        =  0o  cos   >0  sn 


(7) 


dt  dt  ~/p  J-\  p 

(8) 


80 

we  have 

dx  =:  -  —  a0  sin  e  ds 
dy  =  «0  cos  <p0 .  cos 

Then  since 

tl  dg  =  r-  de,  df=  cos  <p  .  gdg,  dJ-  =         ha  = 

ao  r  Tir2  j/£>o 

using  the  values  of  sin  e,  cos  e,  in  terms  of  sin/,  cos/^  we  find 

dx  k  sin/      dy   cos/-|-ee 

^  i/S~'   ^  "~t  i/^T 

ft 
And  these  give 

k  sin/'  dx    i/p<> 

l/p  dz      i/p 

k  cos/  _         dy     VP« ke0 

!/_p~~  rfz      i/p 


VP* 

v.  .  i 

ft 

The  equations  (8)  then  become 


_ 
,    dy  T(1    ,      x  d5  _  I/Pol  =  f( 

dt     dz  LV      ;  d<     j/p  j    j 


n  f 

the  constant  —         being  included  in  the  integral. 


SLU^     #r  )  ^ 

"  dt        dz  L^  '  dt         Vpl  ~J  p 

-to   ,  ft  rn  ,    ^dL  .  .  T/J.-I 


We  will  now  transform  equations  (9),  and  for  this  purpose  we  multiply  the  first 
-~ 


by       ,  the  second  by  -~  ,  and  noting  that 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


81 


we  have 


dv_ 
dt 


_  **f .  Sr\dt  +  si"  /  r ( 
)  j  \ 


Now  multiply  the  first  of  (9)  by  y,  the  second  by  x,  putting  for  J°  its  value 

VP 
given  by  (6),  noting  that 


we  have 


«*  ~  <fy    -  7, 


v)  ^ .  =  I  -     l      f.  Sr  dt  -     y     C(X-  Sin/.  Sr)dt 

dt  fc,/«V  Cp  K/nV  7> 


(11) 


We  can  write  --  in  the  form 

dt 


dz 


dz 
__. 


We  have 


r    '    dt  dz      dt 

2 

Wn  .  -  °.   .  COS  0n,     Of3 


cos 

2 


Making  use  of  these  relations  we  find 


dz  1 


and  for  ,-  given  above  we  have 


^    dz  yp 

v'  'dT 


A.  P.  S. — VOL.  XIX.  K. 


82  A   NEW   METHOD   OF   DETERMINING 

The  equation  (11)  is  thus  changed  into 


dz 


=  f  (l  +  2^)  Srdt- 
J  v  ) 


The  equations  (10)  and  (12)  can  be  put  in  briefer  form. 
Let 


Xs  =  X- sin/  8r,       Yc  =  Y+  cos/+^  8r. 
P  P 


Then 

-  M  4_  8in/  f 

**  +i*:J 


jo.  _ 

~  (13) 


The  values  of  a?,  £/,  found  in  these  equations  we  get  from 


_      _ 
From  the  expressions  for  —  ,    -^  ,  we  have  also 

dz  7    dz 


sin/  __         .         j         (    _ 


(14) 


.  ,_<      +  etc. 

(15) 


The  quantities  given  by  equations  (14)  and  (15)  are  found  in  equations  (13) 
without  the  integral  sign.    They  can  be  put  under  the  sign  of  integration  and  regarded 


THE   GENERAL    PERTURBATIONS    OF    THE   MINOR   PLANETS.  83 

as  constant  if  we  designate  all  magnitudes  in  these  factors  dependent  on  t  by  a  Greek 
letter. 

We  thus  obtain 

d(z  —  0  l         C(-\          O  I  Vo\     o     J*  2         T/-\r  TT     r\  J4 

— ^  r=  —  1(1  -f  2  '  to }  Sr at  —  |  ( X, .  v  —  Ye.t)dt 

at  k |/P(  J  \  \''T>  '  k  vPo  J 

(16) 


These  equations  include  terms  of  the  second  order  with  respect  to  the  mass.     If 
we  put 

W=— 


we  get 

/i/v 

(17) 

J 


In  equations  (17)  </0  is  the  mean  anomaly  for  t  =  0  ;  N  is  the  constant  of  inte- 
gration in  the  value  of  v. 

From  the  value  of  W  given  above,  we  have 


dW 


Now  since 


•      >•   1     d£~\ 

m/.    .T- 

J     r    djj 


-  --  sm.    . 

dr  J     r    dj 


-T^  .         ,     rffl     .  7-     1       rf^l 

Y  =  )sm/.-     hcos/.-.- 

^     dr  J     r     dfj 


84 


A   NEW    METHOD    OF    DETERMINING 


neglecting  the  common  factor  Jc~  (1  +  m), 
we  have 


dt 


19 


VP*\ 


-  cos 


A/i2    .      >T       1  c?i2  7-\  r    ,  fsin/"    f/fi          ,    (cos  /+«,)    dfl     r~l 

4-  ~      ~  I  —  sin  /  4-  cos  /  I  r  +  —  —        .  f  4-  -  .       .  r 

^T/^o  \«*r  f  r  d/          '7  /  ?        AT/P.  L    p       d/1  jj  d/1    ?  J 


And  as 


this  becomes 


v  —  p  sin  0,         ^  =  p  cos 


dt 


-.-  d£ 


df 


4-  2p  cos  Q .  sin  /'.  -v  4-  2p .  —-  cos  o  cos  /'  -)-  2p  . 

«r       r  dj  p    aj 


2p 


oos».co»/dfi,A 


P 


<v 


]*l< 


xdi2    ,    0«  xdfl 

2psm  (/  —  Q)~   4-  2"  cos  (/  —  o)T 

f  ' 


+  2  P-  cos  (/-  Q)      4-  2e0  •    cos  M 


But 


2e0  p-cos  "-     —  2po  •     =  2  1  (e0  p  cos  o  —  po)  =  —  p  .  2 


also 


, 

V/Po 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  85 

Hence  since  &2  (1  +  m)  is  included  in  X,  Y9  72,  S,  we  have 

dw  2  (18) 


dt 

+  2/iop.sin(/—  G>)^- 

If  we  write  h*.  aQ  cos  2<£>0  in  place  of  Jc2  in  equation  (18),  we  have  the  same  ex- 
pression for  -     -  as  that  given  by  HANSEN. 

Equations  (17)  and  (18)  are  fundamental  in  HANSEN'S  method  of  computing  the 
perturbations.     We  will  now  give  HANSEN'S  method  of  deriving  them. 
Using  the  same  notation  as  before,  we  have,  since 

a   1-f-ecos/ 

r  cos  V 

also 

r  cos2  <f>q 

a0  ~  ~  1  -f  ec  cos  /  ' 

hence 

r.a          l+ecos/         cosVo 


r.at  cos  V      *  l-j-P0  cos/'  ' 

Using  f  +  7t0  —  %  in  place  of  f,  and  developing,  we  get 

r-a    .  _  r-{-rcosf.ecos(x  —  7r0)-f  r  sin  /.  e  sin  (/  —  7r0) 

r.ac  ~  o,cos2^^  QA 

Let  us  put 

e  sin  (%  —  7t0)  z=  >7  cos  2<^0, 
ecos  (^  —  7t0)  =: 


since  e  =  sin  <|>,  we  have 

cos  2<?>  —  cos  Vo  (1  —  2e0  ^  —  cos  2^>0  P  •  -  cos  2^>0  »72)- 


86  A   NEW   METHOD    OF    DETERMINING 

With  this  value  of  cos  2<2>> and   r  —  «0 cos >2^0  —  eor  cos/", 
we  find 

r.a      _  a0cosVo — e0.r  cos  f -\-rcosf  (^  cos2  y0-\-e0)-{- r  sin  f.r]  cos 


cos 


at  cos  Vo~Hr  cosy.  I1  cos  Vo~l~?' si11/- ^  cos  Vo 
a0  cos  Vo  ( 1 — 2e0£ — cos  Vo  £2 —  cos  Vo1?'2  j 

and  hence 

1-f-^.— .CO8j^-f-^.—  .  sin/' 


r.a0  1 — 2e0£ — cos'Vo*2 — ( 
From 

dv  _  ^/      _  df     dz 

dl  dt           dz       dt  9 


and 


we  have 


.  COS 


111  like  manner  we  find 


df  «o2 

-  •=.  UQ.  =7  -  COS  d)0- 
az  r 


We  have  therefore 


n.a?,r2.  cos 


(Z<          ^fl.a02-  **'2«  cos 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  87 

If  we  put        —1  +  6,  substitute  the  values  of  — a   .  and  cos  2<2>,  we  get 

"n  *  f  n       ' 


(20) 


(1 — 2e0£ — cos  Vo^2 — cos : 


Further,  in  the  case  of  v,  we  have 


Then  since 


and 


we  have 


If  we  let 


we  find 


1 — 2e?0? — cos  Vo-£2— cos  'Vo-1?2 


cos/.£  +  -  .  sin/. 


=  1  —   e     —  cos          -  -  cos 


h  .  _  (1+6)* 


=  (l  +  5)ir, 


B 


88  A   NEW   METHOD    OF    DETERMINING 

From  the  latter  we  have 


Hence 


__        >   ^      I 

/i         /)„  dt 


If  we  put 


f 

we  have 


We  have  yet  to  express    r°  in  terms  of  the  elements. 


From 
and  from 


COS 


we  have 


_  /  »  \  ^  cos  y>0 

V        /    '  CfOS  <>    ' 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  89 

or 

h   _      an       cos  ?>0 


A0        cos  (p  '    ac 

If  we  put 


cos 


we  have 

7  an 

cos  <p 

These  values  of  h  and  k0  being  substituted  in  the  expressions  for  W,  —  is  found 

Civ 

expressed  in  terms  of  the  elements  and  of  v,  in  a  very  simple  form.     To  find  the  rela- 
tion between  ---  and  v,  we  use  the  equation 

dt 

7~i9! 

(1  +  *)2  = 
and  as  this  is  also  equal  to 


h 
dt 


we  find 

dz          h 


—  (22} 

dt  '  '  h   •'*•«  -^* 


For  the  purpose  of  keeping  the  formulae  simple  and  compact,  HANSEN  makes  use 
of  the  device  of  designating  the  time,  and  the  functions  of  the  time  other  than  the 
elements,  by  different  letters. 

Thus  for  t,  r,  e,  /,  «,  v,  x,  y,  we  write, 

r,  p,  >7,  o,  f,'/?,  £,  v,  respectively. 

Whenever  we  integrate,  these  new  symbols  are  to  be  treated  as  constants,  noting 
that  the  original  symbols  are  used  after  integration. 
A.  P.  s. — VOL.  xix.  L. 


90  A   NEW   METHOD    OF   DETERMINING 

If  in  equation  (21)  we  introduce  <r  instead  of  t  we  shall  have 


where 


.._  coso  .»7._.  sn  a 

„  «o 


ft,  />,  />  «  " 

We  have  also 


The  coordinates  of  a  body  vary  not  only  with  the  time  but  also  with  the  variable 
elements.  In  computations  where  the  elements  are  assumed  constant,  that  part  of  the 
velocity  of  change  in  the  coordinates  arising  from  variable  elements  must,  evidently, 
be  put  equal  to  zero.  Coordinates  which  have  the  property  of  retaining  for  them- 
selves and  for  their  first  differential  coefficients  the  same  form  in  disturbed  as  in  undis- 
turbed motion,  HANSEN  calls  ideal  coordinates. 

If  L  be  a  function  of  ideal  coordinates,  it  can  be  expressed  as  a  function  of  the 
time  and  of  the  constant  elements.  Thus  let  the  time,  as  it  enters  into  quantities 
other  than  the  elements,  be  itself  variable  and,  as  before,  designated  by  t. 

The  function  dependent  on  £,  r,  and  the  elements  we  designate  by  A.     Then 

dL        ~dA 


dt        '    dr 


or 


where  the  superposed  dash  shows  that  after  differentiation  r  is  to  be  changed  into 
Let  us  write  the  equation  (24)  in  the  form 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 

Differentiating  relative  to  r,  we  have 

dPf 


The  differentiation  of  (23)  also  relative  to  t  gives 


_  ,      C 

2~  "  '  '         ' 


dr         />,  '    1/?3  '  dr 


Eliminating  -  by  means  of  (24),  we  have 


•e   h 


dr 


dTF          2/3 
""  r 


JO 

Substituting  in  the  expression  for  J-  we  have 


dW 


Since  ^  is  an  ideal  coordinate,  we  get  from  this 


^T  being  the  constant  of  integration,  and  the  dash  having  the  same  signification  as 
before. 

This  expression  for  v  is  a  transformation  of  that  given  in  the  equation 

1  —  2e,£  —  cos  Vo-£2  —  cos  Vo-1?* 


.     sn 


Since  2;  is  also  an  ideal  coordinate,  we  have  from  (23) 

2  (26) 


being  the  constant  of  integration  and  being  the  mean  anomaly  for  t  =  0. 


92  A   NEW   METHOD    OF    DETERMINING 

When  we  consider  only  terms  of  the  first  order  with  respect  to  the  disturbing 
force,  f  changes  into  r,  and  we  have 

n<p  =  n0t  +  c0  +  n0  (    WQ  dt 

(27) 

_._.        .,    /•  /dW0\    _ 
r  :=  _Zv  —  s  |  (-  - )  a£ 

J  v  czr  y  j 

where 

~\7f7~  f)  *^  *^o  "1        I      O  /*      /^  I      ^i  P        *  /*OQ\ 

and  p  and  o  are  functions  of  T,  being  found  from 

n0  r  +  c0  =  YI  —  eQ  sin  Y} 
p  cos  G)  —  a0  cos  >7  —  aQe0 
p  sin  G)  zz  a0  cos«ji)  0  sin  >?. 

Also  in  the  last  two  terms  of  WQ,  -  is  put  equal  to  unity. 

"o 

When  terms  of  the  order  of  the  square  and  higher  powers  of  the  disturbing  force 
are  considered,  f  cannot  be  changed  into  r.     In  this  case  let 

n0t  — 
Likewise  let 


where 

w^J  is  a  function  of  r  and 

According  to  Taylor's  theorem  we  have 

W=  TTo 


. 

ar  '     dr 

the  value  of  WQ  being  given  by  (28). 


THE   GENERAL   PERTURBATIONS   OF   THE   MINOR  PLANETS.  93 

We  then  have 


dW_  _  dW,  ^  ^       L  .    ^    , 

dS  dr  dr*     <0*          2'      d^'^ 


Retaining  only  terras  of  the  second  order,  the  equations  (25)  and  (26),  replacing 
by  $z,  give 


vz    dt 


dr 

(29) 


The  equation  (26)  has  been  put  in  simpler  form  by  HILL.    For  this  purpose  from  (21) 
and  (22)  we  have 


~ 

dt      ~  dt 


Hence 


Developing  the  second  member  and  adding  W,  we  have 

=  n,  t  +  #o  +  ™o  2  dt.  (30) 


dh   . 

Vb    OLCJJ    IB      IAJ    CA.JJ1COO     _       —    ^ttQ 

dt 
we  find 


The  next  step  is  to  express  "      and  —  in  terms  of  the  disturbing  force.     From  (19) 

dt  dt 


COS 


94  A   NEW   METHOD   OF   DETERMINING 

Using  these  values  of  £  and  vj9  and    e0  p  cos  o  =  «0  cos  2<£>0  —  p,  in  equation  (28),  we 
find 


.  .  , 

ft,  a0  cos  Vo  V*o  cos  Vo 

Since 


T  _     an     _  £|/l-f-m 
"  cos?  " 


we  have  from  the  expression  of  h  already  given, 


dt 

By  means  of 


-  —  1  =1  e  cos  /. 

r 


cos.? 


we  may  transform  the  expressions 


dv          a2 

—  =  —  .  n  cos 

df  r2 


an 


.       /. 

•  e  sm  /, 
cos 


into 


r  -  37  —  ^  —  cos  (f  —  ")  •  he  cos  (#  —  7t0  —  co)  +  sin  (/ — o)  .  Ae  sin  (%  —  n0 

t 

dr  •      /  j?          \     T  /  \  /•  /^~         \     i  / 

—  =  sm  (j  —  co)  .  he  cos  (%  —  7t0  —  o)  —  cos  (/  —  o) .  fte  sm  (^  —  7t0 


THE   GENERAL   PERTURBATIONS    OF   THE  MINOR   PLANETS.  95 

Multiplying  the  first  of  these  equations  by  cos  (f —  G>),  the  second  by  sin  (/ — o>), 
and  adding  the  results,  we  have 

he  cos  (jc  —  7t0  —  G>)  =  (r~  —  h)  cos  (/— o>)  +  ~  sin  (/—«). 

at  ctt 

Substituting  this  value  of  h .  e .  cos  (#  —  7i0  —  o)  in  the  expression  for  TF"0>  noting 
that 


we  have 

-nr^  2.fc../»         ^  cQg    ,-j._      x  r  dv      .  2h9.p         ^  gin    /    -_      s   dr_ 


Differentiating  relative  to  the  time  t  alone,  r  remaining  constant,  and  having  care 
that  all  the  terms  of  the  expressions  be  homogeneous,  we  have 


c^r         /'      civ 
and 

dh  _  F(l+m)       d2u    .  fcV        d2v 


Substituting 


\  p       d*r 

+  »)         r 


96  A   NEW    METHOD   OF   DETERMINING 

we  have 


=  *,     3f-  cos  (  /  -  o)  -  1  +        -      -  [<=os  (  /  -  »)  -1] 

~          1  J 


- 

dt  (    r  h~a0  cos1  $PO  J  *   \  dv 


sn      - 


<30V 


d  W 
This  expression  for  -^  is  the  one  used  by  HANSEN  in  his  Auseinandersetzung. 

It  is  given  in  a  much  simpler  form  in  his  posthumous  memoir,  and  as  the  latter  is  the 
form  in  which  we  will  employ  it,  we  will  now  give  the  process  employed  by  HANSEN 
to  effect  the  transformation. 

Substituting  first  the  value  of  h,  omitting  the  dash  placed  over  certain  quantities, 
noting  that  in  the  posthumous  memoir  <?>  takes  the  place  of  o,  and  remembering  that 
we  are  here  concerned  only  with  terms  of  the  first  order  with  respect  to  the  mass,  we 
have 


/?  \      (  dQ  \ 

—  o)  r(  —  } 

\dr  ) 

From  the  relation 

p  z=  a(l  —  e2)  —  ep  cos  G) 
we  have 

•v 

_  -i          ep  cos  (u 


dW 
An  inspection  of  the  value  of  -^-  shows  that   its   expression  consists  of  three 

parts,  one  independent  of  r,  the  other  two  multiplied  by  p  cos  o,  and  p  sin  G>,  re- 
spectively. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  97 

Put 


=  dB  [+*Y,p  _ 

dt          dt  dt   \a  l  )        dt      a 


V 

tej 


and  we  have 


dS  q       a        f  [~ae  cos  /  ,     e  cos  /     ,        1  ~|  /  cZi2  \    ,  ae  sin  / 

dF               9       «        (Faces/'  ,           (cos/-f-e)  ~|  (  d&  \    ,  a  sin/    „  ^  <?^  \  ) 

**   ~                                                                                                    T 2 \        TV?      J  ~T~  ~   *          V  ~J          /  I    ' 

ndt                   ,/l  _P2  (  L       r                           1 — ez  J  \  df  /  r  \  dr  /  ) 


But 


hence 


o       j   F  a  sin  /     ,  sin/          ~|  /^_\_     a  cos/ 

l  L        r  TU^e^         J\d/  r 


df     _  o?       5  -  "2          ae  cos  /    ,      e  cos  /     , 

:  ^r^  "  "  TT=^)^"  '  '  a-*)1  : 

dr         ae  sin  / 


r          — e 


=  —  a  cos  /; 

^  } 


ds  =  —  3a 


mZ^ 

^F 

ndt 


-  (~  V 


wd<    "  "  y7! — e2         \de  ) 

Again  from 

fd&\   .  .   (d®\    (d£\     ,     (dtt\    fdr\ 
\dg)   '   '   \df)    \dg)          \dr  )    \dg  ) 

A.  P.  S. — VOL.  XIX.  M. 


t/ 


9b  A   NEW    METHOD   OF   DETERMINING 

we  have 


dQ\    _    (dtt\         ?•'  _      (dQ\   resin/ 


J   '       \dcjJ  ayi—e*  \drJ  a(l—e>)  ' 

Eliminating  f—  J  from  the  expression  for  -—  ,  we  have 

dY  2       (a2(l  —  e2)  —  r2      fd&\  r  sin  f  /dQ\ 

-  —  -  J  —  -  -  -  d  (       ]  _l-  •  -     '  —  n.  f  [  _  ] 

ndt~~l—e2\  a*e  \dg  )    raT/l  —  e2         \dr  J 

A    (//  -   fi^1  ^ 

In  the  same  way  we  find  z          -  ' 


dQ\         r~a  cos/       ^  -         e  sin  2/ 


r*  sin/  •        /rfw\         pa  cos/        ^ 

_•_   2^ 


But  if  we  employ  the  relation 


x,/1          ^>2\       I 


r         .    re  cos/ 


fl2\ 
C   y 

,     ,  —  |  ': 

-H"^        A(T 

a  cos/ 


in  the  term,  -       -  Vl  —  e2'A°^  ^ne  preceding  expression,  the  whole  term  becomes 

Crcos/  e  re       ~i          /d&\ 

a  (1—  e2)?        v71"—  e"        a  (1—  e2)d      P  Vdr  / 

Using  the  equation 

0  zr  —  re  cos/  —  r  +  «  (1  —  62)? 
multiplying  by 


e 

CL  T  I  - 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 

adding^  to  the  preceding,  it  becomes 


[- 
~~  La 


rcosf  2e 


Further,  we  have 


dg 


rr 

U  sm/  +  ? 


cos  f 


sn 


"Reducing  this  expression  in  the  same  manner  as  employed  before,  it  becomes 

/'  -i         2  r  cc 
s«)J  ~         a 


d    r~r     ,  r3  sin/  ~i         2  r  cos/-{-  3  a  e 

8  ^ 


a2(l-es) 


Multiply  this  by  cZf/,  the  last  expression  for  —  -  —  becomes 

ttdv 


o 

;j~ 


the  integral  to  be  so  taken  that  it  vanishes  at  the  same  time  with  g. 


„   d3    dY   d>F 
Substituting  these  values  01     -  ,  —  ,  —  ,  m 

ndt  '  ndt  '  nrf^  ' 


dS        dT 


fp  \  ?..• 

=—-  4-  —  (  -  cos  6)  +  f  e  )  -f  —  -  sm 

ndf        nrf^  \a  /        ndta^ 


this  expression  can  be  made  to  take  the  simple  form 


dW 


in  which 


- 


a2(l—  e2)—  r2        2  />  sin  w     /-/2 


99 


J 


1—  e2 


a      1— 


—  e     a 


100 


A    NEW   METHOD    OF    DETERMINING 


Since 


d  .  r2      _        r  sin  / 


a  e  . 


tfig.de 


^ 
cos/> 


we  have 


_^L  r  r^_ 

a2edJ    L  a8  .  rfe 


2(1 


These  expressions  for  A  and  ^  can  be  much  simplified. 
Thus  from 

r2  e4 

-  =  1  +  f  e2  —  (2e  —  J  e3)  cos  #  —  (  J  e2  —  J-  e4)  cos  2#  —  J  e3  cos  3g  —  -  cos  4g  —  etc., 

^JL^J-  *^         tft^VVV**      ^L^'tt^       ^        ^^r-t.       x{!^6^      ^. 

p* 

and  a  similar  expression  for  -,  we  get 


cZ    7*^  /  f>^\  /          ^^\ 

a,e'  d  =  (2  —  -  J  sin  ^  +  (e—  -J 


e  sn 


e3  sin  4^  +  etc., 


/LoT 

»/    La. 


"  • 


—  lain 


r---  sin  4flr  —  etc., 


a2 .  de 


4e  =  —  e  — (2  —  f  e2)  cos  #  —  (  e — f  e3]  cos  2g  —  f  e2  cos  3^  —  |  e3 


cos 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 

From  which  we  obtain 


101 


2e2)cos(y  —  g)        B 
+  (e  +  ^)cos(y  — 20) 

-J  cos  y 

-30) 


-  —  (2 


-(e  +  e4)sin(y— 20) 


cos 


sin  (7  —  Sg) 


+  3  cos  (y  — 40) 


(32) 


Jtjv. — 


These  are  the  expressions  of  A  and  B  whose  values  are  used  in  the  numerical  compu- 
tations. 

When  we  have  the  coefficients  of  the  arguments  in  which  y  is  +  1>  and  — 1,  we 
obtain  the  coefficients  of  the  arguments  in  which  y  is  i  *»  with  very  little  labor. 


d  W 


Let  us  resume  the  expression  for     — =-  ,  that  is, 


ndt 


dW 
ndt 


A  and  B  having  the  values  given  before. 


Since       can  be  put  in  the  form 

a2 


we  have 


a-j/1 — e2        e.dg 


d.- 

•     7  o r  £  '°?  *\d  BW  7 

sm  fc</.        2  -  cos  /  =  —  •  -  =  — -/—   -  cos  Jcq. 
a  de          Z.    de 


A   NEW    METHOD    OF   DETERMINING 


C    I    f      'a"\ 

J  |  (*)" 

But  since 


dR(k)      ' 

*1 


w"  w  -  \e  T  Tse ; cos 

a2 

—  (^e3  —  -^e5)  cos  3^  —  etc. 
we  have 

^7  »(0) 

=  3e. 


Hence  the  integral  just  given  is  simply  --  sin  kg. 
A  and  B  can  then  be  written 

A  =  -  3  +  -L,  f(2  ^  coso  +  3e  }  &=f£±~. 

«2« 


sn 


Putting 


-2  =:  2  jR(K)  cos  A;  y, 


a 


we  have  likewise 


2  ^cos  co  =  — 
a  de 


Introducing  these  values  of  2  -cos  w,  and  2^  sin  w  into  the  expressions  for  ^1  and 
,  after  integration  relative  toj^we  can  write  Win  the  form 


— ~i  —  .          *^>\, 


THE   GENERAL    PERTURBATIONS    OF    THE   MINOR   PLANETS. 


103 


where 


de 


li 


i'g' 


CTand  V  being  two  functions  depending  alone  on  £JL. 
Putting  x  —  +  1,  and  —  1,  we  have 


and  hence 


Thus  we  find 


—   2 


or  putting 


we  have 


de 


de 


,0)  _  «(- 


7T Y  — 

U    —  j  r>(i)      )        r     —  ^(i) 

2^£-  2- 


T 


,d) 


de 


-     de 


de 


de 


de 


'.JL      M&"V 


,(-i) 


_      W  a(D  +  ^W  a(-D 


(33) 


104  A   NEW   METHOD   OF   DETERMINING 

The  values  of  YI(K]  and  O(K)  are  readily  found  from 

")  "K; 

/O   n  1   t$      I         1      ^5\  ^rvc   A/ /I    y^2 1    y>4    _j_    J_  ^o\   f*r*si  *A  f\/1 

~1^/C"""^|,C/      '"     Q  o    (5    I   Ov&    Y 


cos 


We  have 


etc.,  =  etc. 

^^  —3e 

de 


de 


—  (3  #  45    ff 

de     ~—(*e      '  64  e> 


—  (1  p*  4 

—  V3  6   ~5 


de 

etc.  =         etc. 

For  >7(2)  we  have 


^(2) 


=  a*- A  ^  -  -  Tk  e5)  +  a «-  A  ^3  +  3k ^); 

or 

^<2>  =  46-1^-^^^    .  .  _  (34) 

For  0(2)  we  get  at  once 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          105 

In  a  similar  way  we  have 


In  case  of  the  third  coordinate  we  also  compute  the  coefficients  of  the  arguments 
having  no  angle  y  from  those  having  i  y.  For  this  purpose,  putting  K  =  0  in  the 
expression  for  OL(K}  we  have 


£5?rr- 

•  j.    ty  — 


de 


v  de 
where 


For  >7(0)  we  then  have 

Perturbation  of  the   Third  Coordinate. 

Let  5  the  angle  between  the  radius- vector  and  the  fundamental  plane, 

i  the  inclination  of  the  plane  of  the  orbit  to  the  fundamental  plane, 
v  —  <r  the  angular  distance  from  the  ascending  node  to  the  radius-vector. 

"We  have  then 

sin  &  rr  sin  i  sin  (v  —  <r). 

If  we  use  for  i  and  cr  their  values  for  the  epoch  and  call  them  i0  and  &0 ,  £o  being 
the  longitude  of  the  ascending  node,  we  have 

sin  &  =  sin  i0  sin  (v  —  &0)  -f  s  ; 
s  is  the  perturbation. 

Thus  we  find 

s  rr  sin  ?  sin  (v  —  a)  —  sm  iQ  sin  (v  —  Go), 
A.  P.  S. — YOL.  N.  XIX. 


106  A   NEW    METHOD    OP   DETERMINING 

Putting 

p  =.  sin  i  sin  (cr  —  &0)  ,  #  =  sin  i  cos  (cr  —  &0)  —  sin  i0 , 
we  find 

s  =  q  sin  (v  —  &„)  —  p  cos  (v  —  &<>)• 
Instead  of  s,  let  us  use 


r 
U  =—  S 


and  we  have 


u=  ~  q  sin  (»  —  &0)  -      1  p  cos 


Introducing  r  and  calling  R  the  new  function  taking  the  place  of  u,  we  have, 
putting  6)  +  7t0  for  v,  7t0  being  the  longitude  of  the  perihelion,    <U-  ~it~jl  ' 

dE       dq  p      .      ,  dp  p  ,  . 

IF  =  df  ^  Sm  (u  +  ^o  —  So)  —  ^  "^  COS  (co  +  7t0—  So). 

To  find  —  and   —  we  will  employ  the  method  given  by  WATSON  in  the  eighth 

fit  Uv 

chapter  of  his  Theoretical  Astronomy. 

Thus  a  and  (3  being  direction  cosines  we  have 

zl  =  a  x  +  p  y; 

also 

-, 

Zi  =  r  sin  i  sin  (v  —  ff). 
But 

fa  \  _  >($  =.  r  cos  v,  and  y  =  r  sin  v. 

Hence 

3L  =:  —  x  sin  £  sin  G  -}-  y  sin  i  cos  cr? 


THE   GENERAL   PERTURBATIONS    OF    THE   MINOR   PLANETS.  107 

and 

a  =  —  sin  i  sin  cr,      (3  zz  sin  i  cos  cr. 
The  values  of  p  and  q  then  are  given  by  the  equations 

p  —  -  -  a  cos  £0  —  /3  sin  &„, 

2  zz  -  -  a  sin  &o  +  /?  cos  &0  —  sin  ^  ; 

from  which  we  have 

dp  da  dp 

-  =  _coSffi0--sma0-, 

dq  da  dp 

-  =  -8mS0-+oo8ffi0-. 

From  the  equation  zl  zz  a  a?  +  /2  y  we  have,  first  regarding  a  and  /3  as  constant, 
then  regarding  x  and  ^/  as  constant, 

dzA  dx         ~  dy 

dt)  ~~  aW"rP  ^- 

rf^-i  da  dp 


Differentiating  the  first  of  these,  regarding  all  the  quantities  variable,  we  have 

d2*!  _     da   dx        dp  dy  d?x          „  d*y 

~d?~  '  '   dt  ~di  ~~  ~dt  ~dt  ~  '  a  dt*   '  "  ^    df  ' 

^i  being  the  component  of  the  disturbing  force  parallel  to  the  axis  zl9  and  X  and 
!Fthe  other  two  components,  we  have 

Z±  —  a  X  +  j3  Y+  Zcos  i. 
"Writing  for  X  and  Y  their  values 

tfx  .x       dv  ,  70/1       \y 

_  +  ^2(l  +  m);3  ,       -  +  Ar(l  +  m)  -  , 


108  A   NEW    METHOD    OF    DETERMINING 

and  reducing  by  means  of 


we  have 

*•  =  «;£+  0  £  +  *  a  +  •»)  !i  + 


or 


Comparing  this  with  the  other  expression  for  •--*  ,  given  above, 

d  V 

we  have 

da  dx        dp  dy 


From  this  equation,  and  the  value  of  [   _-  j ,  since 

L  '  /  /  J 

dy  dx 


we  find 

da 
dt 

dp 
dt 


=  —  h  r  cos  i  sin  v  Z , 
=  krcos  icosv  Z . 


Substituting  these  values  in  the  expressions  for  ---  and  --- , 


we  have 

dp 
dt 


=  hr  cos  i  sin  (v  — 


dq 

—-  =  hr  cos  i  cos  (v  —  Q>0)  Z . 


THE    GENERAL    PERTURBATIONS   OF    THE   MINOR   PLANETS.  109 

Introducing  these  values  into  the  expression  for  — 

dt 


we  have 


=  7i  r  cos  *  cos  (v  —  &0)  p-  sin  (o>  +  7t0  —  &0)  Z 

at  @ 


-  h  r  cos  i  sin  (v  —  &0)  --  cos  (w  +  ^o  —  So) 

aQ 

h  r  cos  i  --  |sin  G>  cos  (0  —  So  —  (^o  —  So))  I  Z 

a0  L 

-  h  r  cos  *  -     cos  o  sin  (v  —  So  —  (^o  —  So))     - 


li  r  cos  i  p-  sin  (o>  —  f)  —  . 


Introducing  n  =  -  — - —  ,  and  h  zz 

we  have 

==--  sin  (o  —  /  )  a2  -—  cos  i .  (37) 

Let 

1  '     P       •        /  s>\ 

~i/r=?'o  o'sm  ^~~f>> 

then 


cos  *.nc?^ 
To  find  an  expression  for  (7  similar  to  those  for  J.  and  J5  we  have,  first, 

l       rp  r  P  r    •     ^~1 

(7  =  -  sin  G).- cos/ —     coso).     sin/   . 

/l— e2  La0  a  a,  a 


110  A   NEW   METHOD    OF    DETERMINING 

Substituting  the  values  of  r  cos/,   ?  sin /J  given  before,  and   similar  ones  for 
-  cos  a,  -  sin  a,  we  find 


fe  V  afde  )    \a?edg  J  \a<?edb/    \a2de  J ' 

Substituting  the  values  of  these  factors  we  obtain  for  C  the  expression 

O  =.  (1  -  -  -J-  e2)      .sin  (y  -      g) 


-  *  6?)  Bin  (y- 
+  f  e2  sin  (y  — 
-fe8  sin  (  — 


_      d  W  du 

Having  found  the  expressions  for  ——  and 


ndt  ndt .  cos  i 


we  have,  finally,  for  determining  the  perturbations,  the  following  expressions 


n&z  =  n   f  W  dt, 


u          /» 

cos  i        J 


dQ 


dW 


(38) 


Two  integrations  are  needed  to  find  nbz.     We  first  find  W  from  — —  ;  then,  form- 


ndt 


ing  TFand  —  J  —  from  W  we  have  n^z  and  v  by  integrating  these  quantities.     In 


dW 


the  integration  of  — - —  we  give  to  the  constants  of  integration  the  form 


ndt 


&0  +  &!  cos  y  +  &2  sin  y  +  »?(2)  ^i  cos  2  y  -f  >y(2)  fc,  sin  2  y  + 

r  ^ 
I   * 


THE   GENERAL   PERTURBATIONS    OF    THE   MINOR   PLANETS. 


Ill 


Then  in  case  of  —  A      -  we  have 


J  &i  sin  y  -  -  i  &2  cos 


i  -i) 
>?(2)  ^  sin  2  7  -  -  V2)  fc2  cos  2  y  +  ete.  ^  "J 


In  the  second  integration  we  call  the  two  new  constants  C  and  JV^  and  the  con- 
stants of  the  results  are  in  the  forms 

C  +  &0  nt  +  &!  sin  #  —  &2  cos  g  +  %  »?(2)  &i  sin  2  #  -  -  |  >y(2)  &2 


JV~ 


>y       2  cos  2 
—  ^  >7(2)  A?!  cos  2  ^  -  -  J  >?(2)  ^2  sin  2 


,  5. 


In  case  of  the  latitude  the  constants  are    iven  in  the  form 


sin  g 


sn 


cos 


0 


7 


The  constants  are  so  determined  that  the  perturbations  become  zero  for  the  epoch 
of  the  elements.  Hence  also  the  first  differential  coefficients  of  the  perturbations 
relative  to  the  time  are  zero.  We  substitute  the  values  of  g  and  g'  at  the  epoch  in 

ni  fl 

the  expressions  for  n§z.  v,  -    -  ,  -  -   (nbz),  etc.,  including  in  g'  the  long  period  term. 

cos  %     nat 

Putting  the  constants  equal  to  zero,  and  designating  the  values  of  nfiz,  v,  etc.,  at 
the  epoch  by  a  subscript  zero,  we  have  the  following  equations  for  determining  the 
values  of  the  constants  of  integration:  •  «.  13; 


C  +  ^  sin  gr  --  7c2  cos  </  +  i  >?( 


cos  g 


sn 


cos 


cos       — 


cos  2r/  -fetc.     +       (nbz)Q  —  g* 


=  0 


sn 


sin  g  +  I  cos  gr  +  >?(2)     Zi  sin  2^r  +  >y(2)  ^2   cos  2^4-  etc.    +  (~-)0      =  0 

l>  cos  2,7  - 


cos  j,  -  ,  sn  g 


112  A  NEW  METHOD  OF  DETERMINING 

To  find  ki  and  k>.  we  derive  from  the  preceding 


,  [cos<y  —  c  +  >?(2)  cos  2  </  +  57(3)  cos  3  #  H-jetc.J  +  k>  [sin  (j  +  >y(2)  sin  2 


!  [sin  gr  +  2  »7<2>  sin  2  g  +  3  >y(3)  sin  3  g  +)^etc.]  -  -  &2.[cos  g  +  2  >7(2)  cos  2 


The  value  of  N  is  found  further  on. 
Having  Jd  we  find  ^0  from 


-  fc0  —  e  ^  -  3  Z0  +  3  ~-  (nte)o  +  6  (v)0  =  0. 


We  have 

'  2  e 

6()  —       e  L  ,  jY  —        3-  &0          A^i  —  2  A)  ? 


Av^  where  ZQ  is  the  eonstant-  of  W. 

Let  us  find  the  expressions  for  the  constants  N  and  K,  K  being  the  constant  of 
integration  in  the  expression  for  5  -  . 

"0 

The  equation  (22)  we  can  put  in  the  form 


The  differentiation  of  nz  relative  to  the  time  gives 

dz 

— -  —  1  +  &o  +  &o  +  &i  +  periodic  terms, 

Ct6 

where  ZQ  •=.  -  -  32".7162,  in  the  case  of  Althaea,  and  Z^  the  part  to  be  added  when 
terms  of  the  second  order  of  the  disturbing  force  are  taken  into  account. 


THE   GENERAL    PERTURBATIONS    OF    THE   MINOR   PLANETS.  113 

The  expression  for  v  is 

v  •=.  N  -\-  periodic  terms. 
The  approximate  value  of   °  being  1,  the  complete  expression  for  the  integral  of  d  ~ 


is  given  by 


=.  1  +  &3  +  periodic  terms, 


&a  being  the  constant  of  integration. 

Putting  (3^2  —  4z^3  +  etc.)  -     -  2v  (-     -  l)  =  FI  +  periodic  terms,  and  substi- 

it/  \tL  / 

tuting  this  expression,  together  with  those  of  v  and  -  ,  in  the  expression  for  —  ,  we 

/I  dt 

have,  preserving  only  the  constant  terms, 


It  is  necessary  now  to  find  the  value  of  k&  in  terms  of  the  constants.     If  in  the 
expression  for       °  given  by  equation  (18)  we  write  for  p  ,  its  equivalent  «0  cos  2<p0 

Ctt 

-  e0  p  cos  o  ,  we  will  have 

T  1.  1.4  f  £  \  7.2  si~™    ...       ^         .Jf)* 

:}dt 


We  also  have 


Selecting  from  the  expression  for  dWQ  the  terms    not  containing  p  cos  w  and 
p  sin  to,  we  have 


A.  P.  S.  —  VOL.  XIX.  O. 


114  A   ]STE\V    METHOD  OF   DETERMINING 

If  the  eccentric  anomaly  is  taken  as  the  independent  variable  we  have  for  the 
complete  integral 

C  f  7i2    \   /  dQ  \ 

WQ  =.  &0  4-  &!  cos  YI  --  k»  sin  YI  —  7i0   I  [  1  -  -  2rr  )  (  >/•  )  dt. 

0  J  V          *o2;  \VJ  ..& 

• 
Introducing  the  true  anomaly  instead  of  the  eccentric,  we  have, 

cos  <a  -f-  e                                sin  a>  cos  H 
since  cos  >7  = ,      sin  YI  •=. , 

^  1  -4-  e  cos  (a  \-  e  cos 

iA  J  ^  /;2          ^ 

•  -    .        r    '} 

•^ 

1<        /"  C   V  •*  t  C^a  W^ 

Neglecting  the  terms  having  p  cos  o  and  p  sin  G>  we  have  in  TFo  the  constants 
and  e0  h. 

h, 
The  integral  of  d  T  is 

\        '\_  ^ 

Y^ 


From  the  expression  for  d  7°  we  find 


7i 
h 


Integrating  this,  making  use  of  the  value  of  7i° ,  and  adding  the  constants,  we  have 


And  since  the  quantities  under  the  sign  of  integration  do  not  have  any  constant  terms 
we  can  write 

2  -  —  r  =  l  +  &0  +  ekv  +  periodic  terms 


I, 

*• 

7i 


=  1  +  &3  +  periodic  terms 


THE    GEISTEKAL    PERTURBATIONS    OF    THE   MINO^B-PLANETS.  115 

Since  (  °-  -  Ij  is  a  quantity  of  the  order  of  the  disturbing  force  we  have 


from  which  we  get 


Now  putting 

7  O  /  *-! 

(JT-  -1)  -  ~  (  '°-  -l)  rb  etc.  =:  HI  +  periodic  terras, 

substituting  this  expression  and  those  for 

~  h        7)0        /io 
/)„       h   '     ft   ' 


the  preceding  expression  for 

2 

gives,  preserving  only  constant  terms, 


h 


Introducing  this  value  of  ^3  into  the  expression  for  JV  it  becomes 
^=  -  K  4  A;0  4  •  e^  +  3^0)  +  i  (3  ^  +  2/^- 
Preserving  only  the  terms  of  the  first  order  we  have 


To  find  the  value  of  A",  the  constant  of  integration  in  case  of  £  .  -  ,  we  have 

''o 

h  =2  1  +  K  +  periodic  terms, 


116  A   NEW    METHOD    OF   DETERMINING 

also 

'°  —  1  4-  &3  +  periodic  terms. 
h 


From  these  we  get 


j-i  +!'•-!=  A- 

/?  h 


Hence 

K  =  —  \  +H,~l  (fc0  -f  ek,,) 

or,  neglecting  the  term  of  the  second  order, 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  117 


CHAPTER  Y. 

Numerical  Example  Giving  the  Principal  Formula  Needed  in  the  Computation 
Together  with  Directions  for  their  Application. 

ALTH^A  119.  JDPITER. 

g  -  332°   48'    53".2  g'  =  63      5    48.6 

7i  =    11     54    21.1  1  7i'  =  12    36    59.4  ] 

I  I 

£  =  203    51     51.5  }  1894.0  &  =  99    22    59.9  }•  1894.0 

i=      5     44      4.6  J  i'  =    1     18    36.9  J 

$  =      4    36    24.9  $'  =    2    45    57.2 

n  =  855".76428  n'  -  299".12834 

log  n  -  2.9323542  log  n'  -  2.4758576 

log  a  =  0.4117683  log  a'  =  0.7162374 

The  epoch  is  1894  Aug.  23.0. 

The  elements  of  Jupiter  are  those  given  by  HILL  in  his  New  Theory  of  Jupiter 
and  Saturn,  in  which  the  epoch  is  1850.0.  Applying  the  annual  motion  of  57". 9032 
in  7t',  of  36". 36617  in  &',  to  HILL'S  value  of  7t',  and  of  &',  we  have  the  values  given 
above.  The  mass  of  Jupiter  is  TOTT-ST^'  ^he  elements  of  Althsea  are  those  given 
in  the  Berliner  Astronomisches  Jalirbuch  for  1896.  The  ecliptic  and  mean  equinox 
are  for  1890.  To  reduce  from  1890  to  1894  we  employ  the  formula  of  WATSON  in 
his  Theoretical  Astronomy,  pp.  100-102. 

i'  =  i  +  YI  cos  (Q,  —  6) 
Q'  =  Q  +  (t'  —  t)  --  —  YI  sin  (Q—6)  cot . i' 


dl 

t  —  t)       +  YI  sin  (&  —  0) 


A    NEW   METHOD    OF    DETERMINING 

B  =  351°  36'  10"  +  39".79  (t  —  1750)  —  5".21  (f  —  t) 
YI  -  0".46S  (f  —  0 

4*  =  50".246. 

dt 

These  expressions  for  ?'',  &'  and  7t',  can  be  used  for  the  disturbed  body  as  well  as 
for  the  disturbing  body  by  considering  the  unaccented  quantities  to  be  those  given, 
and  the  accented  quantities  those  whose  values  are  to  be  found  for  the  time,  f. 
HARKNESS,  in  his  work,  The  Solar  Parallax  and  Its  Related  Constants,  using  the 

most  recent  data,  gives  the  following  expressions  for  0,  >?,  and  — ,  when  referred  to 

('  u 

1850.0: 

6  =  353°  34'  55"  +  32".655  (t  -  -  1850)  --  8".79  (t  —  t), 
vi  =  0".46654 

dl 
dt 


-  [50".23622  +  0".000220  (t--  1850)] 


Let  an:-, 
n 

we  have  then 

(i  =  0.34955 
2^  =  0.69910 
3^  =  1.04865 
4^  -  1.39820 
5^  =  1.74775 
6^  =  2.09730 
etc.  =.      etc. 
Hence 

1_  S(i  =  —.04865, 
2_  Qu  =  —  .09730. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  119 


This  shows  that  the  arguments  (g  —  3#'),  and  (2g  —  6*7'),  have  coefficients  in  the 
final  expressions  for  the  perturbations  greatly  affected  by  the  factors  of  integration. 
In  case  of  the  argument  (g  —  3g'),  \ve  should  compute  the  coefficients  with  more  deci- 
mals ;  also  those  of  (0  —  3g')  and  (2g  —  3g'),  since  in  the  developments  the  coefficients 
of  these  affect  those  of  (g  —  3</'). 

From 

sin  i  /.  sin  J  (*P  +  4>)  =  sin  J  (&  —  &')  sin  J  (i-ht) 
sin  i-  /.  cos  \  (V  +  <I>)  =  cos  £(&  —  &')  sin  J  (i  —  i') 
cos  i  /.  sin  i  (*P  —  4>)  =  sin  J  (£ 
cos  i  7.  cos  J  (V  —  4>)  =  cos  £(& 

where,  if  &'  >  ^,  we  take  |  (360°  +  Q  —  &'),  instead  of  i  (^  —  ^')5  we  find 

// 
*  =  116°   15'    36.7 

3>=    11     50    33.9 
I-     6    11     35.3 

An  independent  determination  of  these  quantities  is  found  from  the  equations 

cos  2)  sin  q  —  sin  i'  cos  (&  —  Q') 

cos  j9  cos  g  —  cos  i' 

cos  jp  sin  r  =.  cos  i'  sin  (&  —  ^') 

cosjacosf—          cos  (S  —  S') 

sin  jt?  —  sin  i'  sin  (S  —  Q') 

sin  /sin  O  —  sinp 
sin  /cos  $  zr  cos  p  sin  (i  —  q) 
sin  /sin  (1P  —  r)  —  sinj?  cos  (*  —  g) 
sin  /cos  (1P  —  r)  =.  sin  (^  —  g) 

cos  /  =  cos  p  cos  (£  —  g). 


120  A   NEW   METHOD    OF    DETERMINING 

From 

n  =TI  --&  --$> 

IT  =  n'  —  &  —  ip 
we  have 

n  =  156°  11'  55".7  ,  H'  =  156°  58'  22". S. 


Then  from 


k  sin  K  r=  cos  7  sin  II' 
~k  cos  .BT  =  cos  n' 
&t  sin  TTj  •=.  sin  II' 
&i  cos  ^  =  cos  /  cos  II' 


p  sin  P  =  2a2  -  —  2a^  cos  (H 


^  cos  P  =  2a  cos  (|)'  &i  sin  (n  -  -  7 
u  sin  V  =  2a  cos  $  ^  sin  (II  -  -  K) 

v  cos  >"=  2a  cos  <|)  cos  <|>'  A^  cos  (n 

e' 
tw  sin  TF=  ;^  —  2a2  -  sin  P 

e 

t. 

w  cos  TF"=  v  cos  (  F"  —  P) 
Wi  sin  TFi  =  v  sin  (  Tr-  -  P) 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          121 

we  find 

K  =  157°     5'  36".6  log  k  =  9.999614 

Xl  =156    51  7.4  log  ki  =  9.997849 

P   =    93      3  27.0  \ogp  =  9.932748 

V  —  359      6  2.4  log  v  =  0.601463 

TF=266      4  39.5  log  w  =  0.605196 

15  380  log  w,  =  0.601352 


Then  from 

R-  1  +a2  —  2aV%     y2  =  aV2, 

we  have 

log  72  =  0  702855  ,    log  y,  =  7.976024. 

The  values  of  the  quantities  from  II  to  ^2  should  be  found  by  a  duplicate  compu- 
tation without  reference  to  the  former  computation,  since  any  error  in  these  quantities 
will  affect  all  that  follows. 

We  now  divide  the  circumference  into  sixteen  parts  relative  to  the  mean  anomaly, 
and  find  the  corresponding  values  of  the  eccentric  anomaly  E  from 

g  zz  E  —  esin  E  , 

where  e  is  regarded  as  expressed  in  seconds  of  arc.      Substituting  the  sixteen  values 
of  ©in  the  equations 


/sin  (F—  P)  =  w  sin  (E  —  W  )  -- 


we  obtain  the  corresponding  values  of  /'  and  F. 
A.  P.  s.  —  VOL.  xix.  P. 


122  A   NEW    METHOD   OF   DETERMINING 

Then  in  a  similar  manner  from 


\ogq-\ogf-\-y 


where  s  =  206264".8,      logX0  =  9.63778, 
we  find  the  values  of  Q,  (7,  log  q  ,  x  ,  and  y. 
Thus  we  have  found  all  the  quantities  entering  into  the  expression 


Instead  of  this,  we  use  the  transformed  expression 

Q"  =  Nn  (1  +  a2  —  2acos  (E'—Q)}-~>  (1  +  V  —  2&cos  (E'  +  Q))~  "  , 
and  have,  for  finding  the  values  of  JVj  a,  &-,  the  equations 


=.  sm 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          123 


NN_2i      r     (0)        (1)  (2)  n 

'—  O))     '2  :=    16,,   +  &..  cos(£J' — O)+6«  cos2(^'— O)+etc. 

i^y/  i    j*   .**••-**  v  t'  /     '         "'  v  \j  /     l 

L_  7T  IT  -n  _1 


(1)  (2) 

n  cos  (.#'  +  Q)  +  ^  cos2  (^'  +  Q) 

2  2 


+  etc.] 


For  finding  the  values  of  the  coefficients  in  these  expressions  we  use  RUNKLE'S 
Tables  for  Determining  the  Values  of  the  Coefficients  in  the  Perturbatiee  Function  of 
Planetary  Motion,  published  by  the  Smithsonian  Institution.  With  the  sixteen  values 
of  a  as  arguments  we  enter  these  tables  and  find  at  once  the  corresponding  values  of 


(1)          (2)        (3) 


(o)  fe,    6.    6-.  o*      (0)    a3      (i)    a2      (2) 

&  !  ,  then  those  of  -  ,  -2 ,  -" ,  etc.,  etc. ;  -  .  1%  ,&•<>*  >  5  •  *J  ' etc''  etc>> where  P  1S  found 

Y  a     a     a  p        2      P        2      >         2 

a2 

from  the  sixteen  values  of  /52  =  - — -2 . 

A.  ~~      Cv 

Since  &  in  (1  —  2&  cos  (j^r  +  §))  is  very  small  it  will  suffice  to  put 

i^'  =  l      ,!!"=:> 

2  2 

(1)  (1) 

B,   =36,    ^5   =56. 

2"  2" 

Then  from 

(»)  »       (i) 

\      -\-r       T>  r»   •   /~\ 

r,      —    -i-    N      rS 

^n     —    2  M 

2"  2" 


(i)  «        (0 

,  =   t.JV  B 

2  2 


we  have,  in  case  of  ^  (   J, 


124  A    NEW   METHOD   OF   DETERMINING 

and,  for  ,ua2  f?)  , 


(0)  3  (I)  3  (1)  3 

6-3  =JVr          C    =       .#3&cos2  *  =      JT  36 sin 


We  divide  by  8  to  save  division  after  quadrature. 

With  these  values 
values  of  kt ,  A, ,  from 


(0      (0  (*) 

With  these  values  of  cn9  s,,,  and  the  values  of  the  coefficients  &M,  we  find  the 

2"         2"  2 


>'-l)  J'+lK     (1) 

CM    ~T~   I  ^ n 

2"         2"  X    2"  2        '      2 


(&, 

\     7, 

V't- 1 ;  \     (.1 

TT  /         TT 


_(i-l)          -,(i+1)\     (1) 
"%       '     \ 


For  i  —  0,  we  find  A;0  from 

(0)    (0)  (1)  (1) 


—  2    n     w      -     n     n 

2"       2"  2"      2" 


Then  in  case  of  /  f       from 


(c) 


where  m'  is  the  mass  of  the  disturbing  body  and  s  =  206264//8 
and  from 


[i(Q  —  g) 

(s) 

f>  «  =  i  m'  s  a?  Jc(  sin  \i(Q—  g)  —  A*]  , 


(a\3  (c)  *) 

^J  ,  we  find  the  values  of  4t  «  and  4f)  „  for  the  16  different  points  of  the 

circumference,  and  the  various  terms  of  the  series. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  125 

(C)  (8) 

Again,  since  Alt  K ,  AL  K  are  given  in  the  forms 


(c)  (c)  (s) 

Ait  K  —  zO^v  cos  v g  +  2 C«,  „  sin  v g 

(«)  (c)  (s) 

J.^  =  2$>  cos  y#  +  2$,  „  sin  »>gr, 


(c)  («)  (c) 

we  have  the  following  equations  to  find  the  values  of  the  coefficients    Q>,   CJ;r,   o<,^ 
w 


(0.8  )=  F0+  Ys  (£)  =  F0-F8 

/9  1  fh   —    V-4-    V  (  %  \  —    V  -   .  V 

\A.±\J)  —   _t  2  -j-    _i  IQ  VTTT/   —       2          -*•  10- 

^ 

/7ir;\  —   T^_j_y"  •  (  i  \  —   V         V 

(  i.LO)    —    J-T  ~  '    J-15  \TS)   —    -«-7  -"-M 

(0.4)  =  (0.8  )  +  (4.12) 

(1.5)  =  (1.9  )  +  (5.13) 

(2.6)  =  (2.10)  +  (6.14)  '  (0.2)  =  (0.4)  +  (2.6) 

(3.7)  =  (3.111)  +  (7.15)  (1.3)  =  (1.5)  +  (3.7) 

4  (c0  +  2c8)  =  (0.2) 

4(c0-2c8)  =  (1.3) 

4  (c2  +    <%)  =  (0.8)  —  (4.12) 

4  (c2  -     c6)  =  {  [(1.9)  -  (5.13)]  —  [(3.11)  —  (7.15)]  }  cos  45C 

4(s2+    *6)  =  |  [(1.9)  — (5.13)]  +  [(3.11)  — (7.15)]|  cos  45C 

4(s2-     s6)  =  (2.10) -(6.14) 

8c4=  (0.4)  — (2.6) 
8s,-  (1.5)  —  (3.7) 


126  A   NEW    METHOD    OF    DETERMINING 

4  (Cl  +  c7)  =  (f )  +  [(-&)  —  (A)]  cos  45° 
4  (Cl  —  c7)  =  [(i)  —  (yV)]  cos  22°.5  +  [(TT)  —  (A)]  cos  67°.5 
4  (c3  +  c3)  =  (f)  —  [(A)  —  (&)]  cos  45° 

4  (c3  —  c5)  =  [(  |  )  —  (yV )]  sin  22°.5  —  [(fV)  —  (fV)]  sin  67°.5 
4  Oi  +  s7)  =  [(  |  )  +  (yV)]  sin  22°.5  +  [(^-)  +  (1%)]  sin  67°.5 
4  («A  —  «7)  =  [(A)  +  (A)]  cos  45°  +  (T%) 

.        4  (S3  +  «6)  -  [(  i  )  +  (T75 )]  cos  22°.5  -  [(yV)  +  (fV)]  cos  67°.5 
4  (83 -.5)  =  [(A)  +  (A)]  cos  45° -(A) 

The  values  of  c,,  s,,  must  satisfy  the  equation 

(c)  (s) 

Ait  K,  or  Ait  K  =  \  c0  +  CL  cos  g  +  c2  cos  2g  +  etc. 
+  SA  sin  g  -\-  s>  sin  2#  -f  etc. 

i  answering  to  i  in  6H,  and  x  being  any  one  of  the  numbers,  from  0  to  15  inclusive, 

"2 

into  which  the  circumference  is  divided.     We  use  ct,  sv  as  abbreviated  forms  of  C^  „ 

(s)  (c)        (c)        (s) 

CijV,  etc.    Having  found  the  values  of  c,,  «„  from  the  16  different  values  of  JL0,  Aly  A^ 

(C)  (S)  (C)  (8)  /a\  /a\ 

^42,  A27  .  .  .  A9,  Ac,,  both  for  [i  \  ^)  and  ^a2  ^/i  we  have  the  values  of  these  func- 
tions given  by  the  equation 

(c) 


The  values  of  the  most  important  quantities  from  the  eccentric  anomaly  E  to'  ct, 
s0  needed  in  the  expansion  of  ^  f° j  and  pa2  l-J  ,  are  given  in  the  following  tables, 

first  for  ^  (a-\  ,  and  then  for  ^cr  I a  j  ,  when  not  common  to  both. 


THE   GENERAL   PERTURBATIONS   OF   THE   MINOR  PLANETS. 


127 


Values  of  Quantities  in  the  Development  of  /*(-)  and^a'2(-)  . 


I 

9 

^ 

E  +  W 

E  +  W, 

F—  P 

F 

(  0) 

0    1    If 

0  0  0.0 

0    1    II 

266  4  39.5 

O    I    II 

266  15  38.0 

0    1   II 

266  21  17.2 

O    1   II 

359  24  44.2 

(  i) 

24  24  4.2 

i>(.»()  28  43.7 

290  39  42.2 

290  8  7.8 

23  11  34.8 

(  *) 

48  26  37.2 

314  31  16.7 

314  42  15.2 

313  40  58.4 

46  44  25.4 

(  3) 

71  52  24.9 

337  57  4.4 

338  8  2.9 

336  53  39.3 

69  57  6.3 

(  4) 

94  35  14.0 

0  39  53.5 

0  50  52.0 

359  41  1.3 

92  44  28.3 

(  5) 

116  36  51.7 

22  41  31.2 

22  52  29.7 

21  59  7.8 

115  2  34.8 

(  <;) 

138  4  29.4 

44  9  8.9 

44  20  7.4 

43  47  3.8 

136  50  30.8 

(  T) 

159  8  !!>.<; 

65  12  59.1 

65  23  57.6 

65  8  48.4 

158  12  15.4 

(  «) 

180  0  0.0 

86  4  39.5 

86  15  38.0 

86  13  41.4 

179  17  8.4 

(  !>) 

200  51  40.4 

106  56  19.9 

107  7  18.4 

107  15  14.8 

200  18  41.8 

(10) 

221  55  30.6 

128  0  10.1 

128  11  8.6 

128  28  47.5 

221  32  14.5 

(11) 

243  23  8.3 

149  27  47.8 

149  38  46.3 

150  8  27.6 

243  11  54.6 

(12) 

265  24  46.0 

171  29  25.5 

171  40  24.0 

172  23  51.4 

265  27  18.4 

(13) 

288  7  35.1 

194  12  14.6 

194  23  13.1 

195  17  19.4 

288  20  46.4 

(14) 

311  33  22.8 

217  38  2.3 

217  49  0.8 

218  43  0.9 

311  46  27.9 

(15J_ 

335  35  55.8 

241  40  35.3 

241  51  33.8 

242  28  57.5 

335  32  24.5 

V 

1613  47  17.9 

w 

1433  47  18.6 

9 

L°g-/- 

y 

X 

Q 

Log.  q. 

Log.  a 

(  ») 

0.612427 

—.001251 

ii 
-  12.2 

0    1    II 

359  24  32.0 

0.611176 

01706582 

(  i) 

0.612078 

—.000860 

'+431.5 

23  18  46.3 

0.611218 

0.706349 

(  2) 

0.609315 

—.000081 

+598.0 

46  54  23.4 

0.609234 

0.705534 

(  3) 

0.605242 

+.000981 

+390.0 

70  3  36.3 

0.606233 

0.704403 

(  4) 

0.601312 

+.001292 

-  58.6 

92  43  29.7 

0.602604 

0.703241 

(  5) 

0.598569 

+  .000846 

—476.9 

114  54  37.9 

0.599415 

0.702241 

(  6) 

0.597310 

+.000091 

—626.7 

136  40  4.1 

0.597401 

0.701493 

(  ?) 

0.597194 

—.000956 

—435.1 

158  5  0.3 

0.596238 

0.701011 

(  ») 

0.597621 

—.001322 

-  15.7 

179  16  52.7 

0.596299 

0.700788 

(  '•') 

0.598109 

—.000997 

+408.7 

200  25  30.5 

0.597112 

0.700494 

(10) 

0.598532 

—.000152 

+618.1 

221  42  32.6 

0.598380 

0.700021 

(11) 

0.599177 

+.000777 

+496.6 

243  20  11.2 

0.599954 

0.699872 

(12) 

0.600584 

+.001278 

+  96.7 

265  28  55.1 

0.601862 

0.700504 

(13) 

0.603163 

+.001032 

—363.1 

•  288  14  43.3 

0.604195 

0.702020 

(14) 

0.606734 

+.000148 

—600.1 

311  36  27.8 

0.606882 

0.704038 

(15) 

0.610302 

—.000825 

—452.4 

335  24  52.1 

0.609477 

0.705810 

21 

4.823835 

+     3 

—  0.5 

1613  47  17.4 

4.823838 

5.622201 

i? 

4.823834 

2 

0.7 

1433  47  17.9 

4.823842 

5.622200 

128 


A   NEW   METHOD    OF    DETERMINING 


Values  of  Quantities  in  the  Development  of 


and 


9 

X 

Zi 

Log.  b. 

Log.  a. 

a. 

Log.  IT. 

0    /    // 

i   n 

(  °) 

53  23  45.3 

7  57.83 

7.063818 

9.701484 

0.502902 

9.695669 

(  i) 

53  26  41.3 

7  57.78 

7.063792 

9.701945 

0.503437 

9.695880 

(  2) 

53  14  15.6 

7  59.97 

7.065778 

9.699988 

0.501173 

9.695892 

(  3) 

52  54  33.7 

8  3.30 

7.068781 

9.696876 

0.497594 

9.695837 

(  4) 

52  28  55.6 

8  7.35 

7.072405 

9.692804 

0.492951 

9.695616 

(  5) 

52  6  31.2 

8  10.95 

7.075601 

9.689226 

0.488907 

9.695421 

(  6) 

51  53  41.2 

8  13.23 

7.077613 

9.687169 

0.486597 

9.695400 

(  7) 

51  46  50.0 

8  14.55 

7.078774 

9.686068 

0.485364 

9.695430 

(  8) 

51  49  41.2 

8  14.49 

7.078721 

9.686526 

0.485877 

9.695629 

(  9) 

52  0  52.3 

8  13.57 

7.077913 

9.688321 

0.487889 

9.696120 

(10) 

52  18  36.9 

8  12.12 

7.076635 

9.691160 

0.491089 

9.696905 

(11) 

52  36  21.2 

8  10.34 

7.075061 

9.693986 

0.494294 

9.697532 

(12) 

52  49  37.5 

8  8.19 

7.073153 

9.696093 

0.496699 

9.697631 

(13) 

52  58  10.6 

8  5.58 

7.070825 

9.697448 

0.498251 

9.697141 

(14) 

53  5  12.5 

8  2.58 

7.068133 

9.698559 

0.499527 

9.696354 

(15) 

53  13  54.4 

7  59.70 

7.065534 

9.699932 

0.501109 

9.695743 

r 

77.553783 

3.956815 

77.569096 

r 

77.553803 

3.956845 

77.569088 

(0) 

(i) 

(i) 

(0) 

(i) 

(2) 

9 

Log.  !<?! 

Log.  icj 

Log.  fa. 

Log.  ll 

Log.  &t 

Log.  6t 

"2" 

"2 

"2 

2 

2 

f 

(  o) 

8.792579 

6.16064 

4.47527w 

0.332110 

9.748094 

9.329969 

(  i) 

8.792790 

5.98934 

6.02920 

0.332186 

9.748669 

9.331018 

(  2) 

8.792802 

4.98551n 

6.16173 

0.331867 

9.746235 

9.326571 

(  3) 

8.792731 

6.05070n 

5.97267 

0.331369 

9.742375 

9.319511 

(  4) 

8.792526 

6.16734n 

5.14693n 

0.330730 

9.737346 

9.310298 

(  5) 

8.792331 

5.9821  9n 

6.05562n 

0.330182 

9.732946 

9.302224 

(  6) 

8.792310 

4.93934 

6.17378n 

0.329872 

9.730425 

9.297590 

(  1) 

8.792340 

6.03383 

6.01614w 

0.329707 

9.729076 

9.295111 

(  8) 

8.792539 

6.17549 

4.57507^ 

0.329776 

9.729636 

9.296143 

(  9) 

8.793030 

6.05359 

5.99045 

0.320045 

9.731836 

9.300183 

no) 

8.793815 

5.23282 

6.17067 

0.33047> 

9.735322 

9.306586 

(n) 

8.794442 

5.94812n 

6.07618 

0.330914 

9.738805 

9.312970 

(12) 

8.794541 

6.16466^ 

5.36611 

0.331246 

9.741407  i 

9.317738 

(13) 

8.794051 

6.07296/1 

5.94202n 

0.331460 

9.743073 

9.320808 

(14) 

8.793264 

5.23742n 

6.16200n 

0.331637 

9.744461 

9.323327 

(15) 

8.792653 

5.97789 

6.04134?? 

0.331858 

9.746165 

9.326443 

2 

2.647715 

77.912926 

74.508222 

I' 

2.647721 

77.912945 

74.508268 

THE    GENERAL    PERTURBATIONS    OF    THE  MINOR  PLANETS. 


129 


Values  of  Quantities  in  the  Development  of  ^        and 


(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

(9) 

9 

Log.  &j 

Log.  Jj 

Log.  &! 

Log.  ll 

Log.  &J 

Log.  bl 

Log.  &j 

2 

"2" 

1 

2" 

2" 

2 

2 

(  °) 

8.954999 

8.60017 

8.2570 

7.9215 

7.5915 

7.2654 

6.9426 

(  i) 

8.956515 

8.60214 

8.2594 

7.9244 

7.5947' 

7.2691 

6.9468 

(  2) 

8.950082 

8.59373 

8.2490 

7.9120 

7.5804 

7.2528 

6.9286 

(  3) 

8.939865 

8.58036 

8.2326 

7.8926 

7.5578 

7.2271 

6.8997 

(  4) 

8.926521 

8.56292 

8.2110 

7.8668 

7.5280 

7.1932 

6.8617 

(  5) 

8.914818 

8.54760 

8.1921 

7.8444 

7.5020 

7.1636 

6.8285 

(  6) 

8.908100 

8.53882 

8.1812 

7.8314 

7.4870 

7.1466 

6.8094 

(  7) 

8.904506 

8.53411 

8.1754 

7.8244 

7.4789 

7.1373 

6.7991 

(  8) 

8.906000 

8.53606 

8.1778 

7.8273 

7.4822 

7.1411 

6.8033 

(  9) 

8.911861 

8.54373 

8.1872 

7.8386 

7.4953 

7.1561 

6.8201 

(10) 

8.921142 

8.55588 

8.2024 

7.8565 

7.5160 

7.1796 

6.8464 

(11) 

8.930392 

8.56797 

8.2172 

7.8742 

7.5367 

7.2031 

6.8728 

(12) 

8.937298 

8.57701 

8.2285 

7.8875 

7.5520 

7.2205 

6.8923 

(13) 

8.941742 

8.58283 

8.2355 

7.8960 

7.5618 

7.2317 

6.9048 

(14) 

8.945388 

8.58760 

8.2415 

7.9030 

7.5700 

7.2410 

6.9152 

(15) 

8.949898 

8.59349 

8.2488 

7.9117 

7.5800 

7.2524 

6.9280 

2 

71.449530 

68.55219 

65.7484 

63.0060 

60.3071 

57.6402 

54.9995 

I' 

71.449597 

68.55223 

65.7482 

63.0063 

60.3072 

57.6404 

54.9998 

3 

(1)         (i) 

(0) 

(1) 

(2) 

(3) 

9 

Log.i^ 

Log.  -i  c3 

Log.  |  s3 

Log.  %  63 

Log.  63 

Log.  68 

Log.  63 

"2" 

2 

"2". 

2 

2 

2 

(  o) 

8.183917 

5.42374 

3.73837n 

0.280319 

0.417421 

0.200612 

9.961097 

(  i) 

8.184550 

5.25307 

5.29293 

0.281000 

0.418474 

0.202090 

9.963016 

(  2) 

8.184586 

4.24928ft 

5.42550 

0.278120 

0.414013 

0.195824 

9.954877 

(  3) 

8.184421 

5.31430ft 

5.23627 

0.273612 

0.406981 

0.185917 

9.941987 

(  4) 

8.183758 

5.43028n 

4.40987ft 

0.267827 

0.397890 

0.173060 

9.925223 

(  5) 

8.183173 

5.24454n 

5.31797?? 

0.262860 

0.390004 

0.161858 

9.910585 

(  6) 

8.183110 

4.20163 

5.43607w 

0.260054 

0.385513 

0.155458 

9.902210 

(  T) 

8.183200 

5.29621 

5.27852n 

0.258559 

0.383116 

0.152039 

9.897732 

(  8) 

8.183797 

5.43847 

3.83805w 

0.259184 

0.384116 

0.153464 

9.899598 

(  9) 

8.185270 

5.31804 

5.25490 

0.261621 

0.388024 

0.159038 

9.906900 

(10) 

8.187625 

4.49962 

5.43747 

0.265530 

0.394254 

0.167901 

9.918485 

(11) 

8.189506 

5.21681ft 

5.34487 

0.269488 

0.400515 

0.176758 

9.930076 

(12) 

8.189803 

5.43364n 

4.63509 

0.272484 

0.405223 

0.183435 

9.938754 

(13) 

8.188333 

5.34047/1 

5.20953ft 

0.274429 

0.408267 

0.187732 

9.944350 

(14) 

8.185972 

4.50257n 

5.42714ft 

0.276036 

0.410773 

0.191265 

9.948948 

(16) 

8.184139 

5.24121 

5.30466ft 

0.278037 

0.413885 

0.195644 

9.954643 

S 

65.482568 

2.159554 

3.209203 

1.421019 

79.449192 

Z' 

65.482592 

2.159606 

3.209266 

1.421076 

79.449289 

A.  P.  8. — VOL.  XIX.  Q. 


130 


A   NEW   METHOD    OF    DETERMINING 


Values  of  Quantities  in  the  Development  of  l*(~)  and  [*a2(° J 


(4) 

.(5)   1          (6) 

(7) 

(8) 

(9) 

9 

Log-.  &o 

^D     OT 

Log.  63 

Log.  63 

Log.  63 

Log.  63 

Log.  63 

2 

f 

2 

2 

2 

2 

(  o) 

9.70884 

9.4484 

.  9.1822 

8.9118 

8.6383 

8.3621 

(  i) 

9.71121 

9.4512 

9.1854 

8.9155 

8.6423 

8.3665 

(  2) 

9.70116 

9.4393 

9.1716 

8.8998 

.8.6247 

8.3471 

(  3) 

9.68524 

9.4203 

9.1496 

8.8747 

8.5965 

8.3158 

(  4) 

9.66450 

9.3955 

9.1207 

8.8418 

8.5595 

8.2747 

(  5) 

9.64638 

9.3739 

9.0956 

8.8131        8.5273 

8.2389 

(  6) 

9.63600 

9.3614 

9.0813 

8.7968 

8.5089 

8.2184 

(•I) 

9.63043 

9.3549 

9.0735 

8.7880 

8.4991 

8.2077 

(  8) 

9.63276 

9.3576 

9.0766 

8.7914 

8.5030 

8.2119 

(  9) 

9.64181 

9.3684 

9.0893 

8.8058 

8.5191 

8.2298 

(10) 

9.65617 

9.3856 

9.1093 

8.8287 

8.5449 

8.2585 

(11) 

9.67052 

9.4028 

9.1292 

8.8515 

8.5705 

8.2868 

(12) 

9.68125 

9.4156 

9.1440 

8.8684 

8.5893 

8.3078 

(13) 

9.68816 

9.4237 

9.1537 

8.8791 

8.6015 

8.3213 

(14) 

9.69382 

9.4305 

9.1614 

8.8882 

8.6118 

8.3329 

(15) 

9.70087 

9.4389 

9.1711 

8.8992 

8.6240 

8.3464 

2 

77.37450 

75.2339 

73.0471 

70.8269 

68.5804 

66.3134 

I' 

77.37462 

75.2341 

73.0474 

70.8269 

68.5803 

66.3132 

9 

Log.  k0 

Log.  lcl 

Log.  &2 

Log.  &3 

Log.  Jc4 

Log.  7c5 

Log.  k6 

Log.  k7 

(  o) 

8.824187 

8.54492 

8.12562 

7.750420 

7.39550 

7.0523 

6.7168 

6.4105 

(  i) 

8.824302 

8.54433 

8.12588 

7.751220 

7.39678 

7.0540 

6.7190 

6.4054 

(  2) 

8.823605 

8.53875 

8.11916 

7.742693 

7.38634 

7.0416 

6.7046 

6.3714 

(  3) 

8.822665 

8.53172 

8.10982 

7.730361 

7.37091 

7.0232 

6.6832 

6.3298 

(  4) 

8.821701 

8.52543 

8.09963 

7.716100 

7.35261 

7.0007 

6.6565 

6.2932 

(  5) 

8.821143 

8.52236 

8.09246 

7.705215 

7.33807 

6.9826 

6.6349 

6.2764 

(  6) 

8.821183 

8.52300 

8.09009 

7.700585 

7.33130 

6.9737 

6.6239 

6.2809 

(  ?) 

8.821397 

8.52470 

8.08981 

7.699023 

7.32855 

6.9698 

6.6187 

6.2913 

(  8) 

8.821810 

8.52671 

8.09164 

7.701551 

7.33151 

6.9732 

6.6226 

6.3027 

C  9) 

_8.822444 

8.52829 

8.09567 

7.707159 

7.33895 

6.9824 

6.6337 

6.3093 

(10) 

8.823323 

8.52965 

8.10077 

7.715298 

7.35002. 

6.9965 

6.6506 

6.3129 

(11) 

8.824009 

8.53059 

8.10550 

7.723069 

7.3607GI 

7.0100 

6.6669 

6.3147 

(12) 

8.824233 

8.53159 

8.10915 

7.728940 

7.36874 

7.0202 

6.6793 

6.3196 

(13) 

8.824055 

8.53359 

8.11233 

7.733450 

7.37462 

7.0274 

6.6879 

6.3342 

(14) 

8.823809 

8.53721 

8.11622 

7.738311 

7.38053 

7.0345 

6.6960 

6.3608 

(15) 

8.823826 

8.54164 

8.12113 

7.744423 

7.38795 

7.0433 

6.7062 

6.3901 

S 

70.583851 

68.25726 

64.85258 

61.793910 

58.89655 

56.0927 

53.3503 

50.6520 

I' 

70.583841 

68.25722 

64.85260 

61.793920 

58.89653 

56.0926 

53.3505 

50.6512 

THE   GENERAL    PERTURBATIONS   OF    THE    MINOR   PLANETS. 


131 


Values  of  Quantities  in  the  Development  of  fi(^\  and  f.ia2(a\'. 


<7 

Log.  fc8 

Log,  &9 

JT,          K>         JTS         Kt         K.         JZi         K,        -X. 

(  o) 

6.0606 

5.7378 

-  0.6 

-  0.4 

/ 
-  0.3 

-  0.3 

-  0.3 

-  0.3 

—  0.3 

-  0.3 

(  ]) 

6.0636 

5.7413 

.     +20.3 

+  12.9 

+11.4 

+  11.1 

+10.6 

+  10.1 

+  9.5 

+  8.3 

(  2) 

6.0454 

5.7212 

4-27.9 

+  17.8 

+15.6 

+  15.2 

+  14.6 

+  14.0 

+13.5 

+  12.5 

(  •'>) 

6.0178 

5.6904 

+  18.4 

+  11.7 

+10.2 

+  10.0 

+  9.7 

-  9.4 

+  9.2 

+  8.8 

(  4) 

5.9830 

5.6515 

-  2.8 

-  1.8 

-  1.6 

-  1.5 

-  1.5 

-  1.5 

-  1.5 

-  1.5 

(  5) 

5.9541 

5.6191 

—22.7 

—14.5 

-12.7 

—12.0 

-11.8 

-11.6 

—11.4 

-11.0 

(  6) 

5.9391 

5.6019 

—29.8 

—19.0 

-16.7 

-15.7 

-15.3 

—14.9 

-14.5 

—13.7 

(  *) 

5.9316 

5.5934 

—20.7 

-13.2 

—11.6 

—10.9 

-10.5 

—10.1 

-  9.7 

-  8.9 

(  8) 

5.9364 

5.5985 

-  0.7 

-  0.5 

-  0.4 

-  0.4 

-  0.4 

-  0.3 

-  0.3 

0.3 

(   ») 

5.9512 

5.6151 

+  19.3 

+  12.3 

+  10.9 

+  10.2 

+  9.8 

+  9.4 

+  9.0 

+  8.2 

(10) 

5.9737 

5.6405 

4-29.1 

+  18.6 

+  16.4 

+15.3 

+  14.9 

+  14.5 

+  14.1 

+13.3 

(11) 

5.9959 

5.6656 

+23.4 

+  14.9 

+13.1 

+  12.3 

+  12.1 

+11.9 

+  11.7 

+11.3 

(12) 

6.0124 

5.6842 

-  4.5 

+  2.8 

+  2.5 

-  2.4 

+  2.4 

+  2.3 

+  2.3 

+  2.2 

(13) 

6.0251 

5.6968 

-17.0 

-10.8 

—  9.5 

—  8.9 

-  8.8 

—  8.7 

-  8.6 

-  8.4 

(14) 

6.0341 

5.7083 

—28.1 

—17.8 

-15.7 

-14.7 

-14.3 

—13.9 

—13.6 

—13.0 

(15) 

6.0468 

5.7224 

—21.0 

-13.4 

-11.8 

-11.0 

-10.6 

-10.2 

-  9.8 

-  9.0 

JT 

45.3439 

—  -     .5 

—     .3 

—     .2 

+     .3 

—     .3 

w 

45.3441 

0 

.1 

0 

+     .8 

.1 

9 

Log.  Jc0 

Log.  \ 

Log.  k2 

Log.  &3 

Log.  &4 

Log.  &5 

Log.  Jc6 

Log.  £7 

(  °) 

8.465272 

8.60289 

8.38621 

8.14674 

7.89481 

7.6341 

7.3679 

7.0975 

(  i) 

8.466247 

8.60407 

8.38777 

8.14874 

7.89694 

7.6369 

7.3712 

7.1013 

(  2) 

8.462637 

8.59849 

8.38030 

8.13935 

7.88563 

7.6238 

7.3561 

7.0843 

(  3) 

8.457236 

8.59018 

8.36903 

8.12505 

7.86829 

7.6033 

7.3326 

7.0577 

(  4) 

8.450550 

8.58006 

8.35509 

8.10719 

7.84645 

7.5774 

7.3026 

7.0237 

(  5) 

8.445362 

8.57214 

8.34391 

8.09259 

7.82837 

7.5559 

7.2776 

6.9950 

(  6) 

8.443224 

8.56872 

8.33868 

8.08543 

7.81922 

7.5446 

7.2645 

6.9800 

(  "0 

8.442508 

8.56750 

8.33651 

8.08224 

7.81495 

7.5395 

7.2581 

6.9726 

(  8) 

8.444020 

8.56954 

8.33902 

8.08521 

7.81840 

7.5433 

7.2623 

6.9771 

8.444679 

8.57452 

8.34564 

8.09354 

7.82847 

7.5551 

7.2760 

6.9925 

(10) 

8.453274 

8.58206 

8.35573 

8.10632 

7.84401 

7.5734 

7.2971 

7.0165 

(11) 

8.458368 

8.58906 

8.36522 

8.11851 

7.85895 

7.5912 

7.3176 

7.0400 

(12) 

8.461465 

8.59345 

8.37153 

8.12680 

7.86927 

7.6036 

7.3320 

7.0564 

(13) 

8.461922 

8.59532 

8.37468 

8.13126 

7.87506 

7.6105 

7.3405 

7.0660 

(14) 

8.461886 

8.59651 

8.37704 

8.13471 

7.87957 

7.6163 

7.3472 

7.0739 

(15) 

8.462852 

8.59905 

8.38088 

8.13992 

7.88616 

7.6242 

7.3564 

7.0845 

_y 

68.69172 

66.90360 

64.93175 

62.85706 

60.7165 

58.5297 

56.3095 

I' 

68.69184 

66.90364 

64.93185 

62.85719 

60.7166 

58.5300 

56.3096 

132 


A   NEW    METHOD    OF   DETERMINING 


Values  of  Quantities  in  the  Development  of 


and 


9 

Log.  &8  Log.  &y 

J5T,   K,   K,   (Q-g)-^  m-9)-K*  3(Q-g)-K3 

1 

/        /    I                  o    /        a    i   it 

(  °) 

6.8240    6.5478 

—0.1 

—0.1    —0.1     359  25.1     358  49.5     358  13  55.0 

(  i) 

6.8280    6.5522 

+4.4 

+4.4    +4.4      0  28.5      1  24.6      2  14  57.0 

(  2) 

6.8092    6.5317 

+  6.0 

+  6.0    +(>.0      l  26.5      3  31.0      5  27  34.4 

(  3) 

6.7795    6.4988 

+3.9 

+3.9    +3.9      2  15.2      4  55.5      7  30  33.9 

(  4) 

6.7414    6.4566 

—0.6 

—0.6    —0.6      2  46.3      5  28.8      8  12   2.4 

(  5) 

6.7093    6.4209 

47 

—4.7    -4.7      2  47.3      5   3.8      7  26  37.11 

(  6) 

6.6921    6.4016 

—6.2 

—6.2    —6.2      2   9.9      3  39.1      5  16  57.0 

(  7) 

6.6837    6.3923 

—4.3 

—4.3    —4.3      0  55.7      1  23.2      1  56  :5!i.u 

(  8) 

6.6887    6.3976 

—0.2 

—0.2    —0.2    359  17.6    358  34.3    357  51   3.3 

(  9) 

6.7058    6.4165 

+4.0 

_|_4.0    +4.0     357  36.2     355  38.7     353  35  39.5 

(10) 

6.7327    6.4463 

+  6.1 

+  6.1    +6.1    356  13.4    353   6.5    349  51  14.9 

(11) 

6.758!)    6.4752 

+5.0 

+5.0    +5.0     355  26.8     351  25.5     347  17  2<>.4 

(12) 

6.7773    6.4958 

+  1.0 

+  1.0    -1-1.0     355  24.4     350  55.0     346  24  12.8 

(13) 

6.7883    6.5081 

—3.5 

—3.5    —3.5     356   1.7     351  40.2     347  23  40.2 

(14) 

6.7976    6.5187 

—6.0 

—6.0    —6.0     357   4.6     353  30.7     350   5   3.8 

(15) 

6.8093    6.5317 

—4.5 

_4.5    _4.5    358  15.9    356   3.1     353  56  22.5 

V 

54.0630   51.7961 

.0 

.0      .0    1793  47.8              1781  22   3.6 

I' 

54.0628   51.7957 

+  .3 

+  .3    +  .3    1433  47.3               1421  21  59.1 

9 

4«2-< 

l)  —  % 

^(Q-9 

)  —  -^ 

;  6(e-0 

)-^ 

'(  Q  —  9\ 

—  K 

im-9\ 

)  —  A's  9( 

:«-</)- 

-K, 

(  0) 

o 
357 

/ 
38.5 

o 
357 

/ 
3.0 

o 
356 

/ 
27.5 

o 

355 

i 

52.1 

355 

o    / 
16.7 

354 

41.2 

(  1) 

3 

3.9 

3 

53.2 

4 

42.5 

5 

31.8 

6 

21.1 

7 

10.5 

(  2) 

7 

22.4 

9 

17.4 

11 

12.4 

13 

7.3 

15 

2.2 

16 

57.1 

(  3) 

10 

4.4 

12 

38.3 

15 

12.2 

17 

46.0 

20 

19.8 

22 

53.6 

(  4) 

10 

55.5 

13 

39.0 

16 

22.5 

19 

6.0 

21 

49.5 

24 

33.0 

(  5) 

9 

50.6 

12 

15.0 

14 

39.4 

17 

3.9 

19 

28.4 

21 

52.8 

(  6) 

6 

55.9 

8 

35.6 

10 

15.3 

11 

54.9 

13 

34.5 

15 

14.2 

(  7) 

2 

30.9 

8 

5.5 

3 

40.1 

4 

14.7 

4 

49.3 

5 

23.9 

(  8) 

357 

8.0 

356 

24.9 

355 

41.7 

354 

58.6 

354 

15.5 

353 

32.4 

(  9) 

351 

31.8 

349 

27.7 

347 

23.6 

345 

19.5 

343 

15.4 

341 

11.3 

(10) 

346 

34.9 

343 

17.8 

340 

0.7 

336 

43"7 

333 

26.7 

330 

9.6 

(11) 

343 

8.5 

338 

58.9 

334 

49.3 

330 

39.7 

326 

30.1 

322 

20.5 

(12) 

341 

53.2 

337 

22.1 

332 

51.1 

328 

20.0 

323 

48.9 

319 

17.9 

(13) 

343 

7.7 

338 

52.3 

334 

36.9 

330 

21.5 

326 

6.1 

321 

50.7 

(14) 

346 

40.5 

342 

16.6 

339 

52.7 

336 

28.8 

333 

4.9 

329 

41.1 

(15) 

351 

50.4 

349 

44.9 

347 

39.4 

345 

33.8 

343 

28.2 

341 

22.7 

2- 

1744 

6.5 

I' 

1384 

6.0 

THE   GENERAL   PERTURBATIONS   OF   THE  MINOR    PLANETS. 


133 


In  the  expansion  of 


(«) 

(c)                 (s) 

(c)                  («) 

(c)                    (s) 

(C)                    .    (8) 

9 

A 

A,        A, 

-4-2             -a-2 

4              A 

M.%             -#.3 

-^•4             -(*4 

// 

it                n 

n              rr 

//                  // 

//                // 

(  0) 

13.13109 

6.9027  —.0701 

+2.6281  —.0539 

+  1.10745  —.03418 

+.4889  —.0201 

(  i) 

13.13458 

6.8933  -f.0571 

2.6294  +.0647 

1.10917  +.04356 

.4901  +.0262 

(  2) 

13.11352 

6.8033  +.1712 

2.5849  +.1588 

1.08348  +.10356 

.4751   +.0615 

(  3) 

13.08513 

6.6912  -f  .2633 

2.5254  +.2176 

1.04890  +.13827 

.4553  +  -0809 

(  4) 

13.05615 

6.5922  +-3192 

2.4646  +.2364 

1.01333   +.14604 

.4353  +-0840 

(   5) 

13.03939 

6.5457  +.3187 

2.4259  +.2150 

0.99004  +.12935 

.4224  +-(H33 

(   6) 

13.04058 

6.5584  +-2479 

2.4172  +.1543 

0.98367  +.09095 

.4190  +.0509 

(    <) 

13.04700 

6.5880  +.1067 

2.4198  +.0585 

0.98375   +.03339 

.4190  +.0184 

(  ») 

13.05942 

6.6190  —.0816 

2.4317  —.0606 

0.98937  —.03712 

.4218  —.0211 

(   9) 

13.07850 

6.6377  —.2779 

2.4464  —.1863 

0.99667  —.11189 

.4249  —.0633 

(10) 

13.10500 

6.6498  —.4389 

2.4645  —.2979 

1.00593  —.18002 

.4287  —.1023 

(11) 

13.12573 

6.6578  -.5301 

2.4816  —.3742 

1.01487  —.22886 

.4322  —.1310 

(12) 

13.13248 

6.6727  —.5359 

2.4991   —.3995 

1.02497  —.24789 

.4373  —.1431 

(13) 

13.12612 

6.7090  —.4658 

2.5224  —.3693 

1.03984  —.23254 

.4463  —.1354 

(14) 

13.11967 

6.7727  —.3458 

2.5559  —.2907 

1.06142  —.18555 

.4600  —.1090 

(15) 

13.12018 

6.8478  —.2074 

2.5954  —.1791 

1.08668  —.11537 

.4760  —.0683 

y 

104.75791 

53.5708  —.7340 

+  20.0460  —.5531 

8.26962  —.34421 

+3.5661  —.1992 

\v 

104.75663 

53.5705  —.7354 

+20.0463  —.5531 

8.26992  —.34409 

+3.5662  —.1992 

(C)          (8) 

(c)        (s) 

(c)           (s) 

(c)       (s) 

(c>     (s) 

9 

AQ         AQ 

Aj           Arj 

//    // 

II           II 

II             II 

//     // 

//     // 

(  °) 

+.2217  —.0114 

+.1023  —.0063 

+.0505  —.0036 

+  .0226  —.0019 

+.0107  —.0010 

(  i) 

.2223  +.0151 

.1027  +.0085 

.0498  +.0048 

.0226  +.0025 

.0108  +.0014 

(  2) 

.2138  +-0350 

.0978  +.0194 

.0451  +.0105 

.0211  +.0057 

.0099  +.0030 

(  3) 

.2028  +.0454 

.0916  +.0249 

.0401  +.0128 

.0192  +.0071 

.0089  +-0038 

(  4) 

.1916  +.0465 

.0856  +.0252 

.0365  +.0126 

.0176  +.0070 

.0080  +.0037 

(  5) 

.1848  +.0401 

.0821  +.0215 

.0356  +.0109 

.0167  +.0059 

.0076  +.0030 

(  6) 

.1832  +.0277 

.0815  +.0147 

.0368  +.0078 

.0166  +-0040 

.0076  +.0021 

(  7) 

.1833  +.0099 

.0816  +.0052 

.0384  +.0028 

.0168  +.0014 

.0077  +.0007 

(  8) 

.1847  —.0116 

.0823  —.0062 

.0394  —.0035 

.0169  —.0017 

.0078  —.0009 

(  9) 

.I860  —.0346 

.0826  —.0185 

.0388  —.0102 

.0168  —.0051 

.0077  —.0026 

(10) 

.1870  —.0561 

.0827  —.0301 

.0372  —.0160 

.0166  —.0083 

.0075  —.0043 

(11) 

.1880  —.0722 

.0827  —.0389 

.0354  —.0199 

.0163  —.0108 

.0072-—  .0056 

(12) 

.1904  —.0793 

.0837  —.0429 

.0350  —.0216 

.0163  —.0120 

.0072  —.0062 

(13) 

.1956  —.0756 

.0867  —.0411 

.0369  —.0210 

.0173  —.0116 

.0077  —.0060 

.2041  —.0613 

.0918  —.0336 

.0414  —.0180 

.0190  —.0096 

.0087  —.0051 

(15) 

.2140  —.0387 

.0978  —.0214 

.0468  —.0120 

.0210  —.0062 

.0098  —.0033 

S 

+  1.5765  —.1105 

+.7077  —.0598 

+.3219  —.0318 

+.1467  —.0168 

+.0674  —.0087 

i' 

+  1.5768  —.1106 

+.7078  —.0598 

+.3218  —.0318 

+.1467  —.0168 

+.0674  —.0086 

134 


A   NEW   METHOD   OF   DETEKMINING 


In  the  expansion  of  jj.  a2 1  ^ 


(e) 

(C)                         (8) 

(c)                   (8) 

(<0                            («) 

(c)               (») 

9 

A 

A,        A, 

A.^           -42 

A.%          A.% 

A          A, 

it 

n              n 

//             // 

n            n  • 

n             n 

(  o) 

23.3520 

+32.0569  —0.3301 

+  19.4613  —0.4009 

+  11.2092  —0.3464 

+  6.269  —0.258 

(  i) 

23.4045 

32.1423  -j-0.41  99 

19.5273  +0.5272 

11.2569  +0.4603 

6.300  +0.347 

(  2) 

23.2107 

31.7192  +1.0033 

19.1618  +1.2486 

10.9731  +1.0737 

6.096  +0.802 

(  3) 

22.9239 

31.1043  +1.3503 

18.6375  +1.6470 

10.5748  +1.4097 

5.813  +1.041 

(  4) 

22.5737 

30.3821   +1.4503 

18.0367  +1.7240 

10.1342  +1.4580 

5.516  +1.063 

(  5) 

22.3056 

29.8387  +1.2952 

17.5937  +1.5122 

9.8190  +1.2644 

5.310  +0.912 

(  6) 

22.1960 

29.6180  +0.9110 

17.4156  +1.0505 

9.6988  +0.8734 

5.239  +0.626 

(  *) 

22.1595 

29.5473  +0.3342 

17.3564  +0.3782 

9.6618  +0.3118 

5.219  +0.222 

(  8) 

22.2368 

29.6867  —0.3713 

17.4552  -0.4367 

9.7264  —0.3654 

5.259  —0.204 

(  9) 

22.4249 

30.0100  —1.1187 

17.6808  —1.3068 

9.8617  —1.0915 

5.331  —0.786 

(10) 

22.7157 

30.5036  —1.8033 

18.0224  —2.1155 

10.0630  —1.7762 

5.436  —1.285 

(11) 

22.9837 

30.9679  —2.3042 

18.3471  —2.7150 

10.2558  —2.2962 

5.536  —1.667 

(12) 

23.1482 

31.2707  —2.4810 

18.5835  —2.9616 

10.4121  —2.5144 

5.627  —1.839 

(13) 

23.1725 

31.4193  —2.3026 

18.7500  —2.7837 

10.5580  —2.3763 

5.739  —1.748 

(14) 

23.1706 

31.5386  —1.8212 

18.9291  —2.2155 

10.7412  —1.9027 

5.895  —1.409 

(15) 

23.2222 

31.7564  —1.1097 

19.1791  —1.3716 

10.9764  —1.1843 

6.091  —  .882 

r 

182.6038 

246.7758  —3.4423 

147.0656  —4.1071 

82.9580  —3.5000 

+45.337  —2.564 

w 

182.5968 

246.7862  —3.4356 

147.0719  —4.1125 

'  82.9644  —3.4985 

+  45.339  —2.563 

(C)                        (8) 

(C)                        (8) 

(c)                («) 

(C)                         (8) 

(c)              («) 

9 

•4»      ^-5 

A      A 

A           A 

(  °) 

+3.440  —0.177 

+  1.863     —.115 

+1.000  —.072 

+.532  —.044 

+  .282  —.027 

(  i) 

3.458  +0.240 

1.874     +.157 

1.005  +.098 

.535  +.060 

.283  +.036 

(  2) 

3.318  +0.550 

1.781     +.356 

.944  +.221 

.497  +.134 

.260  +.076 

(  3) 

3.130  +0.706 

1.660     +.453 

.868  +.279 

.450  +.167 

.231  +.098 

(  4) 

2.937  +0.713 

1.540     +.453 

.797  +.276 

.409  +.164 

.208  +.095 

(  5) 

2.812  +0.606 

1.467     +.381 

.756  +.232 

.377  +.133 

.196  +.078 

(  6) 

2.772  +0.413 

1.448     +.260 

.748  +.157 

.383  +.092 

.195  +.053 

(  ?) 

2.766  +0.146 

1.446     +.091 

.750  +.055 

.386  +.032 

.197  +.019 

(  8) 

2.789  —0.175 

1.459     —.110 

.757  —.053 

.389  —.039 

.199  —.023 

(  9) 

2.824   —0.522 

1.474     —.329 

.760  —.199 

.389  —.117 

.197  —.067 

(10) 

2.870  —0.855 

1.491     —.540 

.759  —.326 

^.385  —.192 

.193  —.111 

(11) 

2.915  —1.115 

1.505     —.705 

.757  —.425 

.379  —.251 

.187  —.144 

(12) 

2.963  —1.235 

1.528     —.783 

.767  —.473 

.382  —.280 

.188  —.162 

(13) 

3.042  —1.179 

1.582     —.753 

.803  —.457 

.404  —.272 

.201  —.158 

(14) 

3.164  —0.957 

1.670     —.615 

.867  —.378 

.446  —.227 

.227  —.133 

(15) 

3.312  —0.604 

1.775     —.391 

.942  —.243 

.495  —.147 

.259  —.087 

I 

24.253  —1.723 

12.780  —1.094 

+  6.639  —.648 

+3.423  —.392 

+  1.752  —.232 

v/ 

24.259  —1.722 

12.783  —1.095 

6.641  —.660 

+3.415  —.395 

1.751  —.225 

THE    GENERAL   PERTURBATIONS   OF    THE   MINOR    PLANETS. 


135 


.  (0  (s)  (c)  (s) 

The  Quantities  %CttV  ,  JC1>:,  £$>  ,  J$(>  ,  arranged  for  Quadrature  in  the  Expansion  of 


u 


G)- 


<=0 

i-l 

,-,=2 

<  =  8 

<  =  4 

;  =  5 

,•  =  6 

,=, 

(c) 

(c) 

L 

+£[209.51454] 

+53.571 
—.735" 

II 

+20.046 

—.553 

+8.26978 
—.34414 

// 
+3.566 

—.199 

+  1.576 
—.110 

+.707 
—.060 

(7tC> 

^1,1 

+.25653 

+.548 

+.382 

+.22949 

+.129 

+.071 

+.038 

v=l 

(8) 

a!i8) 

—.25027 

+  1.706 
—.122 

+  1.273 
—.046 

+.78997 
—.01129 

+.456 
+.002 

+.253 

+.005 

+.138 
+  .006 

(c) 

+.022 

+.017 

+.00807 

+.003 

+.001 

.000 

(c) 

+.00463 

+.257 

+.096 

+.05847 

+.038 

+.024 

+.013 

«,2} 

—.170 

—.003 

+.01835 

+.017 

+.007 

+  .004 

*_2 

(8) 

+.12279 

+.128 

+.080 

+.04667 

+  .026 

+.015 

+  .001 

(c) 

. 

+.065 

+  .048 

+  .03063 

+.018 

+.010 

+.006 

(c) 

+  .03070 

+.020 

+.007 

+.00662 

+.005 

+.002 

+.001 

(s) 

—.003 

+.002 

+.00216 

+.002 

+.001 

+.001 

v=3- 

(8) 

+  .05945 

+.041 

+.023 

+.01319 

+  .006 

+  .003 

+.002 

(c) 

. 

000 

—.001 

—.00217 

—.002 

—.001 

—.001 

'<£ 

+.00037 

+.001 

+.00030 

<& 

000 

+.00052 

a!f 

+.00055 

000 

+.00076 

gS 

—.001 

—.00103 

136 


A   NEW    METHOD   OP   DETERMINING 


(c)  (s)  (c) 

The  Quantities  |Q>,  ,  \G^V  ,  J$> 


(s) 


rt>  ,  arranged  for  Quadrature,  in  the  Expansion  of 
,/a\8 

(i) 


fcO 

i=i 

(-=2 

i-3 

i=4, 

1=6 

4  =  6 

4*zz  7 

•  Q 

i=9 

f       (0 

n 

V<,0 

n 

+±[364.6002] 

+246.7810 

+  147.068 

+  82.9613 

n 
+45.338 

+24.256 

n 

+  12.781 

+  6.640 

II 

+3.419 

+  1.751 

(c) 
t,0 

—3.4388 

-4.110 

—3.4992 

—2.562 

-1.722 

-1.095 

—.654 

—.392 

—.228 

- 

(c) 

a,i 

+4.3500 

+4.6277 

+3.873 

+2.8862 

+  1.956 

+  1.253 

+.771 

+.461 

+  .270 

+.154 

(s) 

+  7.8438 

+  9.373 

+  7.9505 

+5.816 

+  3.910 

+2.488 

+  1.514 

+  .898 

+.521 

a!r 

-1.8014 

-1.1511 

—.801 

—.3643 

—.106 

+.017 

+.062 

+  .078 

+  .058 

+  .049 

(c) 

; 

+  .1015 

+.104 

+.0731 

+.043 

+.024 

+  .011 

—.008 

+  .003 

+.001 

(c) 

—.2566 

+.0899 

+.294 

+.3888 

+.384 

+.327 

+.252 

+.193 

+  .134 

+.086 

(8) 

+.1010 

+.296 

+.3297 

+.302 

+.239 

+.173 

+  .116 

+.078 

+.047 

<£ 

+  1.1803 

+  1.1209 

+.883 

+.6281 

+.418 

+  .266 

+.162 

+  .093 

+  .058 

+.031 

(c) 
$.2 

L 

+.3367 

+.400 

+  .3459 

+.255 

+.170 

+.106 

+  .065 

+.034 

—.018 

0,8 

+  .1113 

+.1140 

+.099 

+.0809 

+  .066 

+.049 

+.035 

+.024 

+  .013 

+  .012 

Si'* 

—.0170 

.000 

+.0059 

+  .012 

+.015 

+  .015 

+.015 

+  .013 

+.008 

~ 

(s) 

+.5132 

+  .6602 

+.317 

+  .2097 

+.130 

+.076 

+.043 

+  .020 

+.012 

+  .002 

(c) 

—.0138 

—.030 

—.0344 

—.032 

^.027 

—.020 

—.005 

—.010 

—.005 

<c 

+  .0177 

+.0085 

+.003 

+  .0028 

+  .002 

+.002 

.000 

+.001 

.000 

+  .001 

-4 

(s) 

+.0117 

+.005 

+.0061 

+.005 

+.006 

+.004 

+  .004 

+.001 

+  .001 

a!r 

+.0182 

-- 

+.0172 

+.016 

+.0134 

+.010 

+  .006 

+.005 

+.003 

—.002 

+.001 

(c) 

—.0109 

—.022 

—.0182 

—.016 

—.012 

—.008 

+  .002 

—.003 

—.001 

THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


137 


The  quantities  Q,,,  Q,,,  etc.,  of  the  preceding  tables  have  been  divided  by  2  to 
save  division  after  quadrature.  To  check  the  values  of  these  coefficients  we  will  take 
the  point  corresponding  to  g  —  22°.5,  using  the  equation 

(c)  («) 

A!  ,  or  A!  —  I  CQ  +  Ci  cos  g  +  Gz  cos  2#  +  etc. 
+  #1  sin  #  -f-   $>  sin  2y  +  etc., 

noting  that  the  tables  give  one-half  of  the  values  of  these  quantities. 
Thus  we  have 

i=l  i —  2  i—\  i —  2 


c 

i  n 

2^1,0 

+53.571 

n 
+20.046 

(c) 

If! 

2*^1,0    - 

-  0.735 

-  0.553 

(c) 

CM 

+   1.013 

+     .707 

#u  = 

+  1.306 

+     .974 

Cu 

.01)4 

.032 

#u  = 

+     .040 

+     .031 

(c) 

+     .363 

+     .135 

#£  = 

.240 

.004 

•£-1,2 

+     .181 

+     .114 

(c) 
#1,2    = 

+     .092 

+     .070 

(c) 

(s) 

Q,8 

+     .015 

+     .005 

#J,3    - 

.005 

+     .004 

d!s 

+     .077 

+     .043 

#£  = 

0 

.001 

(c) 

(s) 

Ci,4 

0 

.  . 

#1,4    - 

0 

.  . 

rf' 

V/^4 

0 

•  • 

(c) 
#1,4    = 

0 

•  • 

2 

II 
+55.126 

// 
+21.018 

V       

+  0.458 

+  0.521 

P 

+  6.891 

+  2.627 

1.^       

+  0.057 

+  0.065 

^f 

+  6.893 

+  2.629 

£  = 

+  0.057 

+  0.065 

In  this  way  we  check  the  values  of  these  quantities  for  all  values  of  **,  in  case  of 
both  p(5),  and  po»(£). 

Applying  to  the  coefficients  of  the  two  preceding  tables  the  formula 

(a\ n  (°)  («)  (*)  (c)  i- 

2)    =  J22(  a>  =F  #>  )  cos  [(»  =F  v)g  —  iE']  T  JS2(  0,,  dh  #^  )  sin  [(*  T  v)^ 

2  3 

noting  that  J  has  been  applied,  we  have  the  values  of  ^  f  ^ J ,  ^a  Q J    that  follow  : 


A.  P.  s. — VOL.  xix.  R. 


138 


A    NEW   METHOD   OF    DETERMINING 


[l 


G) 


9  E' 

COS 

sin 

1 
COS 

sin 

0       0 

// 

+£[209.51455] 

// 

// 

+£[364.6002] 

// 

1  —  0 

+0.25653 

—0.25027 

+4.3500 

—1.8014 

2  —  0 

+0.00463 

+0.12279 

—0.2566 

+  1.1803 

3  —  0 

+0.03070 

+0.05945 

+0.1113 

+0.5132 

4  —  0 

+0.00037 

+0.00055 

+0.0177 

+0.0182 

—  2  —  1 

+0.023 

—0.041 

+  0.1310 

—0.6464 

—  1  —  1 

+0.427 

—0.193 

—0.0112 

-1.4577 

0  —  1 

-1.158 

+  0.101 

—3.2161 

+  1.0496 

1  —  1 

+53.571 

+0.735 

+  246.7810 

+3.4388 

2  —  1 

+  2.254 

—0.144 

+  12.4716 

-1.2526 

3  —  1 

+0.087 

+  0.063 

+0.1909 

+0.7842 

4  —  1 

+0.016 

+  0.041 

+  0.0970 

+0.6740 

—  1  —  2 

+0.099 

—0.287 

0  —  2 

+0.098 

—0.129 

—0.001 

-1.283 

1  —  2 

—0.891 

+0.029 

—5.500 

+  0.697 

2  —  2 

+20.046 

+0.553 

+  147.068 

+  4.110 

3  —  2 

+  1.656 

—0.063 

+  13.246 

—0.905 

4  —  2 

+0.093 

+0.032 

+0.590 

+0.483 

0  —  3 

+0.00446 

—0.01101 

'    +.0750 

—0.1753 

1  —  3 

+0.04011 

—0.07730 

+.0591 

—0.9741 

2  —  3 

—0.56048 

+0.00322 

—5.0643 

+0.2912 

3  —  3 

+8.26978 

+0.34414 

+  82.9613 

+3.4992 

4—3 

+  1.01947 

—0.01936 

+  10.8367 

—0.4375 

5  —  3 

+0.07682 

+0.01603 

+0.7185 

+0.2822 

6  —  3 

+  0.00879 

+0.01536 

+0.0868 

+0.2441 

1  —  4 

+0.003 

—0.004 

+0.053 

—0.098 

2  —  4 

+0.020 

—0.044 

+  0.082 

—0.674 

3  —  4 

—0.326 

—0.005 

—3.859 

+  0.062 

4  —  4 

+  3.566 

+0.199 

+45.338 

+2.562 

5  —  4 

+0.585 

—0.001 

+  7.772 

—0.149 

6  —  4 

+0.055 

+0.008 

+0.687 

+0.163 

7  —  4 

+0.078 

+0.162 

2  —  5 

+0.005 

+0.045 

H-  0.033 

—0.049 

3  —  5 

+0.016 

—0.025 

+0.088 

—0.095 

4  —  5 

—0.182 

—0.007 

—2.657 

—0.041 

5  —  5 

+  1.576 

+0.110 

+  24.256 

+  1.722 

6  —  5 

+  0.325 

+0.004 

+  5.163 

—0.006 

7  —  5 

+0.031 

+0.004 

+  0.567 

+  0.436 

4  —  6 

+0.009 

—0.008 

+0.079 

—0.269 

5  —  6 

—0.100 

—0.006 

-1.717 

—0.073 

6  —  6 

+0.707 

+0.060 

+  12.781 

+  1.095 

7  —  6 

+0.176 

+  0.005 

+  3.260 

+0.050 

8  —  6 

+0.018 

—0.005 

+0.426 

+0.057 

THE    GENERAL   PERTURBATIONS   OF   THE  MINOR    PLANETS.  139 

We  have  next  to  transform  the  expressions  for  ^  f^J  and  fj.a2  (*~\    just  given 

into  others  in  which  both  the  angles  involved  are  mean  anomalies. 
From 


beginning  with  m  =.  5,  we  find  the  values  of  r5  for  values  of  ef  from  f  to  e'4. 
Then  we  find 


Putting  m  •=.  4,  we  find  the  values  of  r4  as  in  the  case  of  r5.     Then  we  get  p±  from 


(0) 

We  proceed  in  this  way  until  we  finally  have  the  values  of  p{.    Then  we  find  Jh,  ^  or 
from 


2"  J 


(0)  7< 

j  i 72  i  i_ 

/»'   &  ^~~  I       j 

n  2  4 


where  I  — 


(m) 

and  «7^/^  from 


(m)  (0) 

T  ,  f,  =  «7  , 

/l  -2  .1 


The  details  of  the  computation  are  as  follows : 


140 


A   NEW   METHOD    OF    DETERMINING 


Computation  of  the  J  functions. 


*  = 

y 

e' 

1* 

2e' 

2  G 

3e' 

n    f 

^e 

4,6' 

log.? 

8.38251 

8.68354 

8 

.85963 

8.98457 

9 

.08148 

9.16066 

9.22761 

9.28560 

log.  rb 

2 

.31646 

2.01543 

1 

.83934 

1.71440 

1 

.61749 

1.53831 

1.47136 

1.41337 

tog.  #3 

7 

.68354 

7.98457 

8 

.16066 

8.28560 

8 

.38251 

8.46169 

8.52864 

8.58663 

log.  r4 

2 

.21955 

1.91852 

1 

.74243 

1.61749 

1 

.52058 

1.44140 

1.37445 

1.31646 

log.n  —  Iog.jp5 

4 

.53601 

3.93395 

8 

.58177 

3.33189 

3 

.13807 

2.97971 

2.84581 

2.72983 

Zech 

-  1 

—  5 

-12 

-20 

-31 

-45 

-62 

-81 

2 

.21954 

1.91847 

1 

.74231 

1.61729 

1 

.52027 

1.44095 

1.37383 

1.31585 

log.  p4 

7 

.78046 

8.08153 

8 

.25769 

8.38271 

8.47973 

8.55905 

8.62617 

8.68415 

log.  rs 

2 

.09461 

1.79358 

1 

.61749 

1.49255 

1.39564 

1.31646 

1.24951 

1.19152 

Diff. 

4 

.31415 

3.71205 

3 

.35980 

3.10984 

2.91591 

2.75741 

2.62334 

2.50737 

Zech 

—  2 

-9 

-19 

—  34 

-52 

-76 

-103 

-  135 

2 

.09459 

1.79349 

1 

.61730 

1.49221 

1.39512 

1.31570 

1.24848 

1.19017 

tog.  »3 

^j   _M.  o 

7.90541 

8.20651 

8.38270 

8.50779 

8.60488 

8.68430 

8.75152 

8.80983 

log.  r2 

1.91852 

1.61749 

1.44140 

1.31646 

1.21955 

1.14037 

1.07342 

1.01543 

Diff. 

4.01311 

3.41098 

3.05870 

2.80867 

2.61467 

2.45607 

2.32190 

2.20560 

Zech 

—  4 

-17 

—  38 

-67 

—  105 

-152 

-206 

—  269 

1.91848 

1.61732 

1.44102 

'1.31579 

1.21850 

1.13885 

1.07136 

1.01274 

log.  p2 

8.08152 

8.38268 

8.55898 

8.68421 

8.78150 

8.86115 

8.92864 

8.98726 

log.  rt 

1.61749 

1.31646 

1.14037 

1.01543 

0.91852 

0.83934 

0.77239 

0.71440 

Diff 

3.53597 

2.93378 

2.58139 

2.33122 

2.13702 

1.97819 

1.84375 

1.72714 

Zech 

—  13 

-51 

-114 

-202 

—  315 

—  454 

-618 

—  807 

1.61736 

1.31595 

1 

13923 

1.01341 

0 

91537 

0.83480 

0.76621 

0.70633 

log.  pi 

8.38264 

8.68405 

8. 

86077 

8.98659 

9.08463 

9.16520 

9.23379 

9.29367 

log.  Z| 

3.53004 

4.73716 

5: 

43852 

5.93828 

6. 

32592 

6.64264 

6.1)1044 

7.14240 

tog-  7 

2.92798 

4.13210 

4. 

83646 

5.33622 

5. 

72386 

6.04058 

6.30838 

6.54034 

-  tog.  I2 

6.7650271 

7.36708n 

7. 

7l926n 

7.9691471 

8. 

16296n 

8.3213271 

8.45522n 

8.5712071 

Diff 

3. 

83704 

3.23498 

2. 

88280 

2.6321)2 

2. 

43910 

2.28084 

2.14684 

2.03086 

Zech 

-7 

-25 

-57 

-101 

-157 

-227 

-308 

—  402 

log.(—  P  +  -) 

6. 

76495n 

7.3669371 

.?. 

7186971 

7.96813n 

8. 

1613971 

8.3190571 

8.45214n 

8.5671871 

4  / 

3. 

23505 

2.63307 

2. 

28131 

2.03187 

1. 

83861 

1.68095 

1.54786 

1.43282 

Zech 

—  26 

—  101 

—  227 

-401 

—  625 

—  896 

—  1213 

—  1575 

log.  J"(0) 

9. 

99974 

9.99899 

9. 

99773 

9.99599 

9. 

99376 

9.99104 

9.98787 

9.98425 

log.  jpx 

8. 

38264 

8.68405 

8. 

86077 

8.98659 

9. 

08463 

9.16520 

9.23379 

9.29367 

log.  Jw 

8. 

38238 

8.68304 

8. 

85850 

8.98258 

9. 

07838 

9.15624 

9.22166 

9.27792 

tog.  ^ 

8. 

08152 

8.38268 

8. 

55898 

8.68421 

8. 

78150 

8.86115 

8.92864 

8.98726 

log.  <7(2) 

6. 

46390 

7.06572 

7. 

41748 

7.66679 

7. 

85988 

8.01739 

8.15030 

8.26518 

log.  ps 

7. 

90541 

8.20651 

8. 

38270 

8.50779 

8. 

60488 

8.68430 

8.75152 

8.80983 

log.JV 

4. 

36931 

5.27223 

5. 

84)018 

6.17458 

6. 

46476 

6.70169 

6.90182 

7.07501 

lOO".  7?, 

7. 

78046 

.8.08153 

8. 

25769 

8.38271 

8. 

47973 

8.55905 

8.62617 

8.68415 

teg.  J(4) 

2. 

14977 

3.35376 

4. 

05787 

4.55729 

4. 

94449 

5.26074 

5.52799 

5.75916 

THE  GENERAL  PERTURBATIONS  OP  THE  MINOR  PLANETS. 


141 


Noting  that   log.  (e/(0)  —  1)  =:  log.  (—  I2  +  ~),  K  —  ?',    and    I  —  h'X,   we   form 


the  following:  tables  : 


h' 

Log.j[>(«C 

x          1    (1)          1    (2) 

^ 

J3)          !    (4) 
A/V   Log.^«/A/v 

1 

6.7649n 

8.38238 

6.4639 

4.3693 

2.1498 

2 

7.0658/1 

8.38201 

6.7647 

4.9712 

3.0527 

3 

7.2415n 

8.38138 

6.9404 

5.3231 

3.5807 

4 

7.3661/1 

8.38052 

.  7.0647 

5.5725 

3.9551 

5 

7.4624/1 

8.37941 

7.1610 

5.7658 

4.2456 

6 

7.5409n 

8.37809 

7.2392 

5.9235 

4.4826 

7 

7.6070/1 

8.37656 

7.3052 

6.0567 

4.6828 

8 

7.6641n 

8.37483 

7.3621 

6.1719 

4.8562 

i'  h'=  —  2  7i'  —  —  1  ~ti  —  -J-: 

Value  of 

,-/   (h'-V) 
T 

h'  h>* 

h'  -6 

A'=7   ^'=8 

l*:=2   .'=3 

*=*  *=5 

1  4.97l2n  6.4639/t  6.76495/z 

8.38201   6.9404 

5.5725   4.2455 

.... 

....    .... 

2  3.3537/1  4.6703ra  8.68341n 

7.36693n  8.68241 

7.3657   6.0668 

4.7835 

....    .... 

3               6.9410 

8.85913/1  7.71869/r 

8.85764  7.6381 

6.4006 

5.1598     

4               4.9714n 

7.36675   8.98344/1 

7.96813n  8.98147 

7.8413 

6.6588    5.4583 

5 

5.6702/1   7.6393 

9.07949n  8.1614/1 

9.07706 

8.0042    6.8709 

6 

6.1012n 

7.8432   9.15756/1 

8.3190n 

9.15471   8.1402 

7  For  h'  =0, 

6.4176/1  8.0061 

9.22320n 

8.4521w   9.21993 

8  we  have 

6.6689n 

8.1423 

9.27965/1  8.5672/1 

9        8.38251n 

6.8777/1 

8.2594    9.32905n 

In  computing  the  values  of  the  J  functions,  the  lines  headed  Zech  show  that 
addition  or  subtraction  tables  have  been  used.  For  convenience,  (J"(0  - 1)  is  em- 
ployed instead  of  J(0\  its  values  being  found  in  the  line  headed  log.  (—  V  +  -J. 


142  A   NEW   METHOD    OF   DETERMINING 

From  the  expression 


h'  being  the  multiple  of  g',  and  being  constant,  and  i'  being  variable,  we  have 


(A'+l)  9       (V+2) 

'  —  etc. 


Now  for  h'  =  +1?  we  have,  if  we  write  the  angle  in  place  of  the  coefficient, 


((ig  -  g')}  =  1  J«  SS  fa  -  &)  +  |  </A,     «.  (^  _  2^')  +  etc. 


(2)  (3) 

£,  S?  ^  +  JS7')  ~  f  ^'   SS  (^  +  2^0  -  etc. ; 


and  for  7i'  —  —  1,  we  have 


(-2)    Qg       m  ^  (-3) 

(0)  (1) 

4-  1 ,7  *,  c-os  (in  4-  ~K'\  4-  -2-  ,7"    ,  c-os  (in  4-  2 TV'}  4-  oto 

I     i  e/— A'  sm  \fci/     I    •*•  /     I     l  t/— A'  sin  V,  "     i     *iJu*   /  n^  CLO. 

Since 

(—TO)  (TO)  (wt)  (m)  (— m)  (TO) 


the  last  two  expressions  give 


(0)  (1) 

((ig  —  g'))  =  JK,  Z  (ig  —  E')  —  2^  SS  (ig  —  W)  ±  etc. 

(2)  (3) 

—  e/A,  sTnS  (^  +  ^)  —  2e7v  SS  (f>  +  ^')  -  etc., 

(2)  (3) 

((ig  +  g'))  =  —Jy  SS  («^  -  -#')  —  2e7v  SS  (ig  -  %E')  —  etc. 

(0)  (1) 

+  «/v  SS  (^  +  JS?;)  -  2^  ^  (ig  +  2^')  ±  etc. 


THE    GENERAL   PERTURBATIONS    OF    TFIE   MINOR   PLANETS.  143 

And  for  the  particular  case  of  i  =  1,  we  have 

(0)  (1)  (2) 

((ff  -  00)  =  «£'  SS  (  g  -  E')  —  2J«  S£  (  g  -  2JS7')  +  3J«  %  (  g  -  3JB")  =F  etc. 

<2)  (3)  (4) 

-  Jv  SS($r  +  E')-2J«  Sg?  (#  +  2^')  -  3</A,  SS  (<7  +  3^')  -etc. 

(2)  (3) 

((g  + 


(0)  (1)  (2) 

SS  (  flr  +  2E')  +  3«7V  sc&s  (  g  +  3^')  =F  etc. 


(0)  (0) 

Instead  of  J^,  ,  we  use  (JK>  —  1),  as  has  been  noted. 
If  we  put  hf  =  +  2,  we  have 

(1)  (0)  (-1) 

((ig  —  2g'))  -  I  Jw  SS  (ig  —.#')  +  f  ,72V  ^  (^  —  2^')  +  f  e/w  SS  (^  -  3-ET')  +  etc. 

(3)  (4) 

'  ^  +  2J0')  -  etc. 


-  ' 


- 

In  the  table  giving  the  values  of  -  JK>>    ,  we  have,  under  h'  =  2,  which  applies  to 

//' 

the  equation  just  given, 

(1)  (3) 

for  i'  =  1,  log.  i<72A,  =  8.38201          log.  (—  i  J^,)  =  4.9712^; 

(0)  (4) 

for  i  -  2,        log.  (     %Jw  —  l)  =  7.36693w        log.  (—  f  <72A  )  =  3.3537^  ; 

(i) 
for  tf  =  3,        log.  (—  f  Jw        )  =  8.85913/i  etc.       =     etc. 

etc.,  etc.  =      etc. 

(3)  (4)    ^ 

We  find  the  values  of  —  J  J^'?  —  f  «Sw  in  the  table  under  7^'  =  —  2.    We  see  that 

i'      (h'-i') 

these  are  the  forms  of  the  function     Jk,^    when  h  —  —  2,  and  i'  =:  1  and  i'  =  2. 


In  the  expansion  of  the  coefficient  of  (ig  —  h'g')  indicated  above  by  ((tg  —  &'ff'))t 
we  have  coefficients  of  angles  of  the  form  (ig  +  i'E').  These  can  readily  be  put  into 
the  form  (  —  ig  —  i'  E'\  but  the  form  employed  is  convenient  in  the  transformation. 


144 


Arranging  the  functions  fi  (a\  ^0?  (a-\   in  this  form,  we  have 

Log.  na-  Log. 


9    E' 

COS 

sin 

COS 

sin 

0  —  1 

0.0637ft 

9.0043 

0.5074ft 

0.0210 

0  —  2 

8.9912 

9.1106ft 

7.0000ft 

0.1082ft 

0  —  3 

7.6493 

8.0418ft 

8.8751 

9.2437ft 

1  +  1 

9.6304 

9.2856 

8.0493 

0.1637 

1  —  1 

1.72893 

9.8663 

2,3923 

0.5364 

1  —  2 

9.9499ft 

8.4624 

0.7404ft 

9.8432 

1  —3 

8.6032 

8.8882ft 

8.7716 

9.9886ft 

1  —  4 

7.4771 

7.6021ft 

8.7243 

8.9912* 

2+  1 

8.3617 

8.6128 

9.1173 

9.8105 

2  -  1 

0.3530 

9.1584ft 

1.0959 

0.0978ft 

2  —  2 

1.30203 

9.7427 

2.1675 

0.6138 

2  —  3 

9.7486ft 

7.5079 

0.7045ft 

9.4642 

2  —  4 

8.3010 

8.6435ft 

8.9138 

9.8287ft 

2  —  5 

6.6990 

7.6532 

3-1 

8.9395 

8.7993 

9.2808 

9.8944 

3-2 

0.2191 

8.7993ft 

1.1221 

9.9566n 

3  —  3 

0.91750 

9.5368 

1.9189 

0.5440 

3  —  4 

9.5132ft 

7.6990ft 

0.5865ft 

8.7924 

3  —  5 

8.2041 

8.3979ft 

8.9445 

8.9777ft 

4  —  1 

8.2041 

8.6128 

8.9868 

9.8287 

4  —  2 

8.9685 

8.5051 

9.7709 

9.6839 

4  —  3 

0.0082 

8.2869ft 

1.0348 

9.6410ft 

4  —  4 

0.5522 

9.2989 

•  1.6565 

0.4085 

4  —  5 

9.2601ft 

7.8451ft 

0.4244ft 

8.6128ft 

4  —  6 

7.9542 

7.9093ft 

8.8976 

9.4298ft 

5-3 

8.8855 

8.2049 

9.8564  ' 

9.4506 

5  —  4 

9.7672 

7.0000ft 

0.8905 

0.1732ft 

5  —  5 

0.1976 

9.0414 

1.3848 

0.2360 

5  —  6 

9.0000n 

7.7782ft 

<j  0.2347ft 

8.8633ft 

6  —  3 

7.9440 

8.1864 

8.9385 

9.3876 

6  —  4 

8.7404 

7.9031 

9.8370 

9.2122 

6  —  5 

9.5119 

7.6021 

0.7129 

7.7782ft 

6  —  6 

9.8494 

8.7782 

1.1066 

0.0394 

6  —  7 

0.0224ft 

8.8451ft 

7  —  6 

* 

0.5132 

8.6990 

7  —  7 

0.8222 

9.8156 

7  —  8 

9.7973ft 

8.7924ft 

THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


145 


We  will  now  give  examples  to  illustrate  the  application  of  the  tables  for  trans- 
forming from  eccentric  to  mean  anomaly,  in  case  of  the  function 


For  the  angle    3g  —  3g'. 

„•/      (h'-i') 


cos 


sn 


(K  =  3) 


Log.  Product. 


Product. 


3  — 

1 

8.9395 

8.7993 

6.9404 

5.8799 

5.7397 

+  .00008 

+  .00005 

3  — 

2 

0.2191 

8.7993ft 

8.68241 

8.9015 

7.6817ri 

+  .07970 

-  .00303 

3  — 

3 

0.91750 

7.5368 

7.71869n 

8.6362ft 

5.2555ft 

-  .04327 

-  .00180 

3  — 

4 

9.5132ft 

7.6990ft 

8.98344ft 

8.4966 

6.6824 

+  .03139 

+  .00048 

3  - 

5 

8.2041 

8.3979ft 

7.6393 

5.8434 

6.0372ft 

+  .00007 

-  .00011 

+8.26978 

+0.34414 

+8.33775 

+0.33973 

1  —  1 


1.72893 
9.6304 


9.8663 
9.2856 


For  the  angle    g  —  og'. 

(h'  =  0) 

8.38251w       0.11144w     8.2488n 
8.38251™       8.0129^       7.6681ft 


—1.29259 

-  .01030 

+0.25653 


—  .01773 

-  .00466 

—0.25027 


—1.04636     —0.27266 


For  the  angle   g  -f-  g'. 


1  —  1 


1.7289 


9.8663 


'  =  -  1) 

6.4639ft 


8.1928ft     6.3302ft 


—  .016 
+0.427 


.000 
+0.193 


+0.411         +0.193 


A.  P.  8. — VOL.  XIX.  S. 


146  A   NEW    METHOD    OF    DETERMINING 

For  the  angle    og  —  og'. 

0  —  1  0.0637n  8.3825w          8.4462  ...       -f       .02794 

+  104.75727 


+  104.78521 


For  the  angles  represented  by  (ig  —  #')>  there  may  be  cases  when  there  are  sensi- 
ble terms  arising  from  g  +  12',  g  +  2.E7',  etc. ;  if  so,  we  use  the  column  for  li'  =  -  -  1, 
and  apply  the  proper  numbers  of  this  column  to  the  coefficients  of  the  angles  named. 
Likewise  in  the  case  of  (ig  -f  #'),  there  may  be  terms  arising  from  the  product  of  the 
numbers  in  the  column  li'  =  1  and  the  coefficients  of  the  angles  g  -f  E',  etc.  This 
will  be  made  clear  by  an  inspection  of  the  two  expressions 


(0)  (1) 

((ig  -  g'}}  -        J»  £  (ig  -  -  E')  -  2<7A  £  (ig  -  -  2E')  ±  etc. 

(2)  (3) 

ig  -W)--  etc., 


(2)  (3) 

((ig  +  flO)  =  -  J«  S  (*  —  E')  —  2,7A,  SS  (ig  -  2E')  --  etc. 

(0)  (1) 

+  J»  SS  (ig  +  -#')  -  2e/v  SS  (^  +  2J7')  ±  etc. 


where  ((ig  —  #')),  ((ig  +  ^'))  represent  not  the  angles  but  their  coefficients. 

In  retaining  the  form  (ig  +  i'E')  instead  of  the  form  (  —  ig  --  i'E')  we  can  per- 
form the  operations  indicated  without  any  change  of  sign  in  case  of  the  sine  terms. 

Making  the  transformations  as  indicated  above,  we  obtain  the  following  expres- 

sions for  the  functions      -     and 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


147 


9    9' 

COS 

sin 

COS 

sin 

0  —  0 

// 

+  104.78521 

// 

// 

+182.3777 

// 

1  —  0                              1.04636 

—0.27266 

1.6046 

-1.9194 

2  —  0                              0.05031 

+0.12527 

0.5606 

+  1.1949 

3  —  0 

-f     0.02860 

+0.05793 

+     0.1067 

+0.4943 

—2  —  1 

—     0.1274 

—0.6468 

—  1  —  1 

+     0.411 

—0.193 

0.0830 

-1.4558 

0  —  1 

1.162 

+0.107 

3.2141 

+  1.1107 

1  —  1 

+  53.583 

+0.734 

+246.9027 

+3.4023 

2  —  1 

+     1.286 

—0.171 

+     5.3656 

-1.4496 

3  —  1 

+     0.014 

+0.066 

0.3758 

+0.8304 

0  —  2 

+     0.070 

—0.127 

—     0.085 

-1.242 

1  —  2 

+     0.399 

+0.053 

+     0.456 

+0.848 

2  —  2 

+  20.093 

+0.551 

4-147.392 

+4.049 

8  —  2 

+     1.056 

—0.086 

+     7.214 

—1.137 

4  —  2 

+     0.027 

+0.033 

0.086 

+0.537 

0-3 

+     0.00815 

—0.01707 

+     0.0718 

—0.2352 

1  —  3 

+     0.04342 

—0.07447 

+     0.0041 

—0.9231 

2  —  3 

0.40733 

+0.03392 

+     2.0442 

+0.5514 

3  —  3 

-f     8.338 

+0.340 

+  83.537 

+3.432 

4  —  3 

+     0.675 

—0.036 

6.432 

—0.659 

5  —  3 

+     0.028 

+0.010 

+     0.079 

+0.449 

2  —  4 

+     0.027 

—0.043 

+     0.050 

—0.637 

3  —  4 

0.275 

+0.023 

+     2.174 

+2.592 

4  —  4 

+     3.628 

+0.197 

+  46.016 

+2.512 

5  —  4 

+     0.397 

—0.013 

4.828 

—0.323 

6  —  4 

+     0.021 

+0.008 

+     0.156 

+0.188 

3  —  5 

+     0.020 

—0.023 

+     0.080 

—0.074 

4  —  5 

+     0.167 

+0.012 

+     1.762 

+  0.241 

5  —  5 

+     1.623 

+0.109 

-  24.829 

+  1.565 

6  —  5 

+     0.224 

—0.004 

+     3.306 

—0.148 

4  —  6 

+     0.012 

—0.008 

+     0.077 

—0.250 

5  —  6 

+     0.092 

+0.007 

4.535 

+0.150 

6  —  6 

+     0.731 

+0.059 

+  13.312 

+  1.085 

148 


A   NEW    METHOD    OF   DETERMINING 


The  transformation  should  be  carefully  checked  by  being  done  in  duplicate,  or 
better  by  putting  the  angle  ig  =  0,  in  all  the  divisions  of  the  two  functions,  having 
thus  only  the  angles  (0  —  E'\  (  0—  2E'),  (0  —  3E'),  etc.,  etc. ;  also  (0  —  g'\  (0  — 
20r'),  etc.  Adding  the  coefficients  in  each  division  of  the  functions  before  and  after 
transformation,  and  operating  on  the  sums  before  transformation  as  on  single  members 
of  the  sums,  the  results  should  agree  with  the  sums  of  the  divisions  of  the  transfor- 
mations given  above. 

The  transformations  of  these  functions  were  checked  by  being  done  in  duplicate, 
but  we  will  give  the  check  in  case  of  another  planet.  We  have  for  the  logarithms  of 
the  sums  before  transformation,  and  for  the  sums  after  transformation  the  following : 


9 

y 

COS 

sin 

g 

g' 

o  - 

-  1  i 

.85407 

1.62090n 

0- 

-  1 

0  - 

-  2    1 

.25778 

1.51473n 

0  - 

—  2 

0  - 

-  3    9 

.7024,* 

1.26993n 

0  - 

-3 

0  - 

-4    0 

.TlOln 

0.9147n 

0- 

-  4 

0  - 

-5    0 

.6632n 

0.3899n 

0  - 

-5 

0  - 

-  6    0 

.4387n 

9.0934 

0  - 

-  6 

0  - 

-  7    0 

.1222w 

9.8069 

0  - 

-  7 

0  - 

-  8    9 

.5965n 

9.8865 

0  - 

-  8 

For  the  angle 

(0-1), 

(0 

-2), 

a 
0.041   -\ 

0.024 

+  1.722 

1 

.007 

•  0. 

873   H 

1.578 

.042 

+ 

.076 

• 

000 

0016 

4-   .037 

+  1 

.346 

4-71. 

462 

-41.774 

.012 
4-  18.104 

—  32 

.019 
.714 

+  70. 

4-  70. 

548 
573   - 

-  40.188 
-40.196 

4-  19.809 
4-  19.811 

-32 
-32 

.318 
.319 

COS 

II 

4-  70.548 

4-  19.809 

4-    0.906 

4.540 

4.707 

3.059 

0.623 

—    0.071 


sm 

-  40.188 

-  32.318 

-  19.352 

9.263 

3.313 

0.330 

4-    0.739 

4-    0.615 

0  —  3. 


+    .062  .037 

-j-    .871  1.574 

4-    .003  4-      .097 

.  -f-    .494  4-      .791 

.020  .011 

.504  -  18.618 

4-  0.906  +  19.352 

_|_  0.902  —  19.355 


The  numbers  in  the  last  line  of  each  case  are  the  sums  of  the  divisions  after  con- 
version when  ig  is  put  =  0. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.  149 

To  have  close  agreement  it  is  necessary  that  all  sensible  terms  in  the  expansion  of 
and  ^a2Qj  be  retained.     In  the  expressions  for  these  functions  given  a  large 

number  of  terms  and  some  groups  of  terms  have  been  omitted  as  they  produce  no 
terms  in  the  final  results  of  sufficient  magnitude  to  be  retained. 

In  transforming  a  series  it  will  be  convenient  to  have  the  values  of  the  (/functions 
on  a  separate  slip  of  paper,  so  that  by  folding  the  slip  vertically  we  can  form  the  pro- 
ducts at  once  without  writing  the  separate  factors. 

o 

The  numerical  expressions  for  ^(^)  and  ^a-2HJ    being  known,  we  need  next  to 

have  those  designated  by  (H)  and  (Z),  which  represent  the  action  of  the  disturbing 
body  on  the  Sun. 

To  find  (H)  we  use  two  methods  to  serve  as  checks.     We  have  first 


(//)  =  i[A7iyi'  +  fc'SA']  cos  (g  -  00          -  J[%/  +  ZyA']  sin  (g  -  cf) 

<£ 

+  i  [ftyiyi'  -  $1  cos  (-  g  -  00  ©  i  [Z«V/  -  ZVA']  sin  (-  g  -  g') 

+  i  fyoyl  cos  (—  Sf)  ~  ipM'  sin  (—  00 

2'  +  ft'SAI  cos  (g  —  2g')        -  2[M1y2'  +  /'yAl  sin  (0  —  20') 

'  --  ^A']  cos  (—^  —  2^)  -h  2[Z^2'  —  7yA']  sin  (—  cj 

+  2  7iW./  cos  (—  2</')         —  2?yA'  sin  ( 


f  [^iy8'  +          '    cos       -  3<         -  *'   sin 


+  etc. 
@ 

where 

(0)  (2)  (0)  (2) 

(1)  (3)  (1)  (3). 


(2)  (4)  (2)  (4) 

3A       -   ^3A    ]  ^3    =    IK*        +    «/3A    Jj 


and  similar  expressions  for  y/,  ^',  y.2',  52',  etc.  ;  noting  that  y0  =  —  3e. 


150  A    NEW   METHOD    OF    DETERMINING 

The  other  expression  for  (H  )  is 

(H)  =  #hyi'  -  -  h'K]  cos  (—  E  —  g')  +  i[Zyi'  -  -  Z'V]  sin  (—  E  —  g') 
+  it  V  +  W]  cos  (J0  -  g)         -  jpy/  +  I'W  sin  (^  ~  </') 
-  ehyi  cos  (—  </')  +  eZ'V  sin  (  —  #') 

4.  2[fty2'  —  /*'&>']  cos  (—  E—  2g')  +  2[fy2'  -  -  Z'^']  sin  (—  E  —  2g 


cos        —     r  -         2  sn        — 


cos  (—  2g')  +  ieZ'^'  sin  (—  2gr') 

+  etc.  +  etc. 

In  both  expressions  for  (H)  we  have 
£  =£&cos(n  —  K) 
h'  =  t  cos  <}>  cos  <?>'  ^  cos  (n  —  JKi)  zz 

—  JT) 
^  sin  (n  —  K)  -  I 


where  as  before 

•v 

ft  =  —  .  206264."8    and    a  =  -. 

I  -f-m  a 

In  the  second  expression  the  eccentric  angle  of  the  disturbed  body  appears  and  we 
must  transform  the  expression  into  one  in  which  both  angles  are  mean  anomalies. 
With  the  eccentricity,  e,  of  the  disturbed  body  we  compute  the  J  functions  just  as 
we  did  in  case  of  e'  of  the  disturbing  body. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


151 


We  have  in  case  of  Althaea 


(0) 


Log.G/-!) 
Log.  J 
Log.  J 
Log.  J 
Log.  J 


\ 

Log.  J 


(4) 


¥ 

e 

f« 

2e 

7.20740w 

7.80894n 

8.16025n 

8.40890n 

9.99930 

9.99719 

9.99368 

9.98872 

8.60344 

8.90341 

9.07774 

9.20016 

6.90632 

7.5077 

7.8587 

8.1068 

5.0329 

5.9356 

6.4630 

6.8365 

3.0347 

4.2384 

4.9418 

5.4403 

(h-i) 


From  these  values  we  may  form  a  table  of  --  J^    as  was  done  for  the  disturbing 

body.     The  values  of  these  quantities  can  be  checked  by  means  of  the  tables  found 
in  ENGELMANN'S  edition  of  BESSEL'S  Werke,  Band  I,  pp.  103-109. 

Finding  the  numerical  value  of  (H)  first  by  the  second  expression,  we  get 


E 

9' 

COS 

a 

sin 

1  - 

i 

+48.154 

+0.651 

-i  - 

i 

+  0.188 

—0.102 

o  - 

i 

-  3.884 

—0.044 

i- 

2 

+  4.644 

+0.062 

-i  - 

2 

+  0.018 

—0.010 

0  - 

2 

-  0.374 

—0.004 

1  - 

3 

+  0.37800 

+0.00510 

1  - 

3 

+  0.00141 

—0.00081 

0  — 

3 

-  0.03048 

—0.00036 

To  transform  we  change  from  (hE — i'g'}  into  (i'g1  —  TiN).  Making  the  transfor- 
mation, writing  also  the  values  found  from  the  first  expression  for  the  sake  of  compari- 
son, and  the  value  of  (I)  which  will  next  be  determined,  we  have 


152 


A   NEW    METHOD    OF    DETERMINING 


(I) 


9     9' 

0  —  1 

COS 

II 

-  5.826 

sin 

a 
—0.066 

COS 

// 
-  5.824 

sin 

a 

—0.066 

sin               cos 

//                   // 

+4.799             +2.043 

0  —  2 

-  0.560 

—0.006 

-  0.562 

—0.006 

+0.463             +0.197 

0  —  3 

-  0.04566 

—0.00057 

-  0.04575 

+0.038             +0.016 

-1  —  1 

+  0.149 

—0.103 

+  0.180 

—0.103 

1  —  1 

+48.076 

+0.650 

+48.079 

+  0.650 

1  —  2 

+  4.631 

+0.062 

+  4.605 

+0.062 

1  -3 

+  0.37740 

+0.00502 

+  0.37738 

+0.00510 

2  —  1 

+  1.927 

+0.026 

+  1.927 

+0.030 

2  —  2 

+  0.186 

+0.002 

+  0.186 

+0.002 

2  —  3 

+  0.011 

0.000 

+  0.015 

0.000 

To  find  the  numerical  value  of  (/)  needed  in  case  of  the  function  a2  (   -),  we  have 

\(LZ/ 

(I)  =       &«'!  sin  (-  •   tf)  +      6y x  cos  (-    g') 
+  4  65'2  sin  (—  20r')  +  4  6y  a  cos  (—  2gf) 
+  9  M's  sin  (— 3gr')  +  9  6ys  cos  (— 


+  etc. 


+  etc. 


where 


b  =  —  ~  cos  d>'  sin  J  cos  II',  &'  =  ^  sin  I  sin  II'. 


Having  the  values  of  ^  Q),  ^a2  (**-}  ,  (^),  and  (/),  we  next  find  those  of 


r\ 

all, 


j      <> 

-r-,         and  a2  —  , 

dr  dz7 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          153 

from 


where 


(1)  (2)  (3) 

e2  -  -  ^t7A   cos  #  —  JJk  cos  2g  —  |  J"3A  cos  3g  —  etc. 


o?n  T  T'  (0>  (2>  Q)  <3) 

-,  sin  (/'  +  IT)  =  —  [<7A,  +  «7V  ]  G!  sin^'  -  -  \\JW  +  ,72X,]  Cl  sin  2^'  —  etc. 


(0)  (2)  (1)  (3) 

-f  |e'c2  —  [J^/  —  «7A'  ]  c2  cos  (/'  —  J  [t^A/  —  J^'  ]  c2  cos  2^  —  etc. 


and  C2  being  given  by  the  equations 


sin/          ,  .          T-T, 
=  --  COS  d)   COS  II 

a 


sin/    .     T-J. 
2  =          smll'. 

a 


We  find 


_I2^]  -  [9.5769400]—  2  [8.38238]  cos  g'  —  2  [6.46366]cos  2g'—  etc. 

+  2  [7.99450]  cos0  +  2  [6.29667]  cos2^-f-  etc. 
_  si^"  ^  sin  (  /  '  +  n')  =    [7.18046]     +  2  [8.39074]  sin  #'  +  2  [6.77809]  sin  2#' 

CC         Of 

—  2  [8.01941]  cos#'  —  2  [6.40668]  cos  20' 


A.  P.  8.  —  VOL.  XIX.  T. 


154  A    NEW   METHOD    OF    DETERMINING 

In  multiplying  two  trigonometric  series  together,  called  by  HANSEN  mechanical 
multiplication, 

let    GCA  the  coefficients  of  the  angles  /\.x  in  case  of  the  sine, 
fin  those  of  the  angles  fix  in  case  of  the  cosine, 
/„  those  of  the  angles  vy  in  case  of  the  sine, 
and     Sp  those  of  the  angles  py  in  case  of  the  cosine. 

The  following  cases  then  occur : 

aA  sin  /la?  .  $p  cos  py  rz  |  a^p  sin  (%x  -\-  py)  +  JaA<^>  sin  tyx  —  py) 
ft.  cos  fix  .  yv  sin  vy  —  \  (3^,  sin  (fix  +  vy)  —  i  (3^  sin  (px  —  vy) 
^  cos  fix  .  $  p  cos  py  =  J  j3^p  cos  (fix  +  p?/)  +  i  ftA  cos  (^x  —  py) 
aA  sin  %x  .  yv  sin  vy  =  —  J  aA/v  cos  (/(x  +  vy)  4~  2  ®<Kyv  cos  (/la?  —  vy). 

In  every  term  of  the  second  members  the  factor  \  occurs.     Hence  before  multiplying 
we  resolve  the  coefficients  of  one  of  the  factors  into  two  terms,  one  of  which  is  2, 

Performing  the  operations  indicated,  we  have  the  values  of  «n,  ar      ,  a2          that 
follow : 


THE   GENERAL   PERTURBATIONS    OF   THE   MINOR  PLANETS. 


155 


ar\ 


dr 


g 

g'                cos 

sin                       cos 

sin 

COS 

sin 

0 

o 

+  104.78521 

" 

+  16.5202 

" 

+0.2828 

•  " 

1 

—  0 

1.04636 

—.27266 

-  2.4398 

-  .6940 

—2.6311 

+  6.0177 

2 

-  0 

.05031 

+.12527 

.3040 

+  .3928 

-  .059 

+  .239 

3 

-  0 

+       .02860 

+.05793 

+     .0274 

+  .1494 

-  .017 

—  .017 

—  1 

—  1 

.231 

—.090 

.431 

—  .355 

.000 

—  .129 

0 

-  1 

+     4.662 

+.173 

-  1.166 

+  .481 

-1.743 

—4.157 

1 

-  1 

5.504 

+.084 

+  18.839 

+  .190 

+  .318 

+  .068 

2 

-  1                       .641 

—.201 

-  1.652 

-  .577 

-k596 

+3.580 

3 

-  1 

+       .014 

+.066 

.240 

+  .288 

-  .059 

+  .232 

0 

2 

+       .632 

—.121 

+     .497 

-  .414 

-  .020 

—  .149 

1 

2 

4.206 

—.009 

-  9.136 

+  .200 

—2.474 

—6.095 

2 

—  2 

+  19.907 

+.549 

+45.566 

+  1.270 

+  .095 

—  .067 

3 

2 

+     1.056 

—.086 

+  1.642 

-  .441 

-  .922 

+2.011 

4 

2 

+       .027 

+.033 

.115 

+  .180 

-  .064 

+  .194 

0 

-3 

+       .05390 

—.01764 

+     .0718 

—  .0602 

—  .030 

+  .017 

1 

O 

.33396 

—.07957 

.4443 

-  .3306 

—  .045 

—  .166 

2 

-  3 

+       .39221 

+.03380 

-  2.1788 

+  .1339 

—1.424 

—3.658 

3 

-3 

+     8.338 

+.340 

+27.227 

+  1.087 

—  .064 

—  .134 

4 

—  3 

.675 

—.036 

+  1.796 

-  .269 

—  .519 

+  1.099 

5 

•  —  o 

+       .028 

+.016 

+     .043 

+  .157 

—  .042 

+  .123 

2 

—  4 

+       .027 

—.043 

—     .054 

—  .210 

—  .046 

—  .146 

3 

-  4 

.275 

+.023 

.880 

+  .908 

—  .784 

—2.078 

4 

—  4 

+     3.628 

+.197 

+  15.430 

+  .882 

—  .038 

—  .106 

5 

4 

.397 

—.013 

+     .883 

-  .137 

—  .282 

+  .586 

6 

-  4 

+       .021 

+.008 

.013 

+  .063 

—  .031 

+  .083 

3 

-5 

+       .020 

—.023 

.034 

—  .078 

+  .020 

—  .130 

4 

—  5 

.167 

+.012 

.281 

+  .044 

—  .411 

—1.150 

5 

-5 

1.623 

+.109 

+  8.605 

-  .543 

+  .024 

—  .227 

6 

-  5 

+       .224 

—.004 

+  1.061 

+  .064 

—  .158 

+  .311 

4 

—  6 

+     0.012 

—.008 

—  0.075 

—0.095 

5 

—  6 

.092 

+.007 

—  2.225 

+  .026 

6 

-  6 

+       .731 

+.059 

+  4.559 

+  .386 

156  A   NEW    METHOD    OF    DETERMINING 

Having  all  we  differentiate  relative  to  ft.  and  obtain  ac  "*. 

dg 

We  then  form  the  three  products,  A.a   "\   B  .  ar  (  '  --  ),   C  .  ar(  -     j  .    To  this  end 

ilg  \dr  /  \dz  / 

we  find  A,  B,  Cy  from 

^  =  —  3  +  2[2  +  ea]cos(y-  g)         B  =  -2  [1+-  f]  sin  (y  -  g) 
+  2  [f  +  f]  cos  (y  —  2g)  -2  [|  +  f]  sin  (y-20) 


(y  —  Sg)  -  2  |e2  sin  (y  - 

^-2f  cos  (/ 
-j-  etc.  -  etc. 


+  etc. 

The  numerical  values  of  Ay  B,  G  in  case  of  Althsea  are 

A=-3  }           ) 

+  2  [0.302429]  cos  (7  —    g)  B  =  --  2  [0.001399]  sin  (7  -       0) 

+  2  [8.604489]  cos  (7  -  -  2g)  -  2  [8.604489]  sin  (y-  -  20) 

-  2  [9.304508]  cos  7  -  2  [8.606234]  sin  r 

+  2  [7.2076]       cos  (7  —  30)  -  2  [7.3836]      sin  (7  -  -  30) 

O  =  +  2  [9.697567]  sin  (7  -      0) 

+  2  [8.30066]  sin  (7  -  -  20) 

-  2  [8.77953]  sin  7 

+  2  [7.08265]  sin  (7  — 


THE    GENERAL   PERTURBATIONS   OF    THE   MINOR    PLANETS. 

For  the  three  products  we  then  have 


157 


A       (d& 

A-a(j; 


o.  «'- 


7 

9 

9' 

sin 

COS 

sin 

COS 

sin 

COS 

„ 

„ 

a 

„ 

,, 

II 

1 

0 

-  0 

+  2.1035 

—0.5371 

+  1.1341 

—0.6804 

—1.3464 

—3.0038 

1 

i 

-  0 

.012 

+  .565 

.4021 

+  .3723 

+  .1287 

+  .2411 

-1 

i 

-  0 

.2530 

+  .0439 

—32.9502 

+  .0549 

—  .3877 

-  .4802 

1 

2 

-  0 

.192 

+  .299 

.0153 

-  .1657 

-  .0049 

+  .0228 

-1 

2 

-  0 

+  2.079 

-  .597 

-  1.1310 

+  .6821 

+  1.2995 

+2.9772 

—  1 

3 

-  0 

+  .261 

+  .457 

.1263 

-  .3720 

+  .083 

+  .2404 

1 

.  2 

I 

+  .462 

+  .181 

+  .432 

—  .348 

—  .076 

+  .243 

1 

—  1 

I 

.266 

—  .015 

+  .453 

+  .461 

—1.881 

f4.454 

1 

0 

I 

—10.992 

+  .153 

—18.335 

+  .187 

+  .354 

-  .642 

—  1 

0 

I 

+  .462 

+  .181 

.477 

-  .349 

—  .228 

+  .572 

1 

1 

—  1 

-  3.680 

-  .815 

+  .929 

-  .559 

-  .815 

-1.785 

—1 

1 

-  1 

+  1.119 

—  .013 

.449 

—  .476 

+  1.906 

—4.470 

1 

2 

-  1 

—  .342 

+  .477 

+  .306 

+  .276 

+  .067 

+  .098 

l 

2 

I 

—11.301 

+  .249 

+  18.336 

-  .188 

-  .178 

—  .359 

—  1 

3 

—  1 

+  2.360 

—  .843 

—  .929 

+  .559 

+  .785 

+  1.760 

—  1 

4 

-  1 

.033 

+  .381 

—  .264 

-  .276 

1 

—  1 

—  2 

+  .232 

.000 

—  .232 

—  .060 

+  .194 

1 

0 

2 

+  6.837 

+  .026 

+  7.300 

+  .235 

-1.230 

+3.029 

-1 

0 

2 

.... 

.... 

—  .001 

+  .009 

1 

1 

2 

—80.684 

+2.195 

—45.412 

+  1.264 

+  .178 

—  .371 

-1 

1 

—  2 

—  .848 

+  .002 

+  .132 

-  .406 

—  .139 

+  .290 

1 

2 

-  2 

+  1.633 

-  .735 

-  3.470 

-  .384 

—  .467 

—1.010 

l 

2 

-  2 

+  16.433 

—  .240 

-  7.317 

-  .235 

+  1.239 

—3.036 

1 

3 

2 

.422 

+  .316 

'+  .048 

+  .168 

+  .024 

+  .023 

l 

3 

2 

—79.078 

+  2.254 

+  45.412 

-1.264 

—  .053 

—  .273 

1 

4 

2 

-  7.937 

—  .500 

.213 

+  .384 

+  .454 

+  .981 

—  1 

5 

2 

.408 

+  .255 

+  .198 

-  .163 

1 

0 

-3 

+  .5985 

—  .1553 

+  .4644 

—  .3261 

—  .0482 

+  .157 

1 

1 

-3 

-  2.6517 

+  .1927 

-  1.1042 

+  .1641 

-  .7083 

+  1.8160 

l 

1 

-3 

—  .0661 

+  .0161 

+  .0541 

+  .0737 

+  .0123 

+  .0180 

1 

2 

-  3 

—50.140 

-[-  1.905 

—27.2994 

+  1.0854 

+  .043 

-  .174 

—  1 

2 

-  3 

+  .828 

—  .1733 

.5308 

+  .3287 

—  .062 

+  .136 

1 

3 

3 

.380 

—  .492 

-  2.8964 

-  .2201 

—  .256 

—  .558 

-1 

3 

—  3 

+  3.482 

—  .073 

—  1.1112 

—  .1645 

+  .707 

—1.818 

1 

4 

-  3 

+  .263 

+  .190 

.115 

+  .147 

+  .010 

+  .005 

—  1 

4 

-  3 

—49.676 

+2.079 

+27.299 

—1.083 

+  .029 

-  .206 

-1 

5 

—  3 

—  6.395 

—  .264 

+  3.899 

+  .217 

+  .257 

+  .534 

158 


dQ 
— 
dg 


c.  <t. 


7 

9    9' 

sin 

COS 

sin 

COS 

sin 

COS 

// 

// 

n 

// 

// 

II 

i 

1  —  4 

—    .165 

—  .170 

.... 

—  .038 

+  .115 

i 

2  —  4 

—  2.229 

+  .187 

+     .264 

+  .939               —  .389 

+  1.029 

—  1 

2  —  4 

-f     .011 

+  .017 

.... 

+  .008 

-  .014 

1 

3  —  4 

—29.032 

+  1.564 

—15.481 

+  .915                +  .022 

—  .083 

—  1 

3  —  4 

+     .058 

-  .187 

—     .089 

+  .175                 -  .024 

+  .051 

i 

4  —  4 

-  1.063 

-  .287 

-  1.504 

-  .098                  -  .140 

—  .300 

—  1 

4  —  4 

+   1.268 

-  .024 

—     .022 

-  .938                +  .390 

—1.033 

—  1 

5  —  4 

—28.751 

+  1.597 

+  15.479 

-  .915                +  .033 

-  .129 

—  i 

6  —  4 

—  4.543 

-  .108 

+  1.506 

+  .098 

1 

2  —  5 

—     .160 

—  .136 

+  .002 

+  .088 

1 

3  —  5 

-  1.654 

+  .132 

—     .063 

+  .063 

—  .206 

+  .570 

—  1 

3  —  5 

+     .012 

-  .014 

.001 

—  .003 

+  .001 

+  .008 

i 

4  —  5 

—16.185 

4-1.082 

-  8.661 

+  .544 

-  .034 

-  .038 

-1 

4  —  5 

+     .015 

-  .148 

.045 

+  .076 

—  .035 

+  .004 

1 

5  —  5 

—  1.061 

-  .158 

-  1.412 

—  .036 

—  .080 

-  .168 

—  i 

5  —  5 

+     .294 

-  .017 

+     .062 

—  .063 

+  .206 

-  .563 

—  1 

6  —  5 

—16.038 

+1.100 

+  8.661 

-  .544 

1 

3  —  6 

—    .121 

—  .063 

i 

4  —  6 

—  1.088 

+  .086 

+  2.052 

+  .038 

1 

5  —  6 

-  8.707 

+  .703 

-  4.516 

+  .387 

-i 

7  —  6 

—  8.818 

+  .711 

+  4.516 

—  .387 

Next  from 


dW 

ndt 


we  find  the  value  of .     Then  we  find  W  and  — —    from 

ndt  cosi 


dW 
ndt 


COS* 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          159 

We  first  form  a  table  giving  the  integrating  factors.     From  log.  n'  zz  2.4758576, 
log.  n  =  2.9323542,  we  have  -  =  0.34954524. 


i  i' 

*+*'*' 

n 

Log.  (<+#£) 

/  1  M 

•  i  -/ft' 

Loaf,  \i-\-i'  —  ) 
\    ft  / 

/   1  \ 

g'(i+i~) 

i  i' 

ft 

-Log-la+i'g) 

—2  —  1 

—2.34954 

0.37098ft 

9.62902ft 

3  —  3 

+  1.95136 

0.29034 

9.70966 

—  l  —  i 

—1.34954 

0.13018ft 

9.86982ft 

4  —  3 

+2.95136 

0.47002 

9.52998 

0  —  1 

—  .34954 

9.54350ra 

0.45650ft 

5  —  3 

+3.95136 

0.5968 

9.4032 

i  —  l 

+  .65045 

9.813217 

0.186783 

1  —  4 

—  .398181 

9.60008ft 

0.39992n 

2  —  1 

+  1.65045 

0.21760 

9.78240 

2  —  4 

+  .601819 

9.77946 

0.22054 

3  —  1 

+2.65045 

0.4233 

9.5767 

3  —  4 

+1.601819 

0.20461 

9.79539 

4  —  1 

+3.65045 

0.5624 

9.4376 

4  —  4 

+2.601819 

0.41528 

9.58472 

—1  —  2 

—1.69909 

0.23021ft 

9.76979ft 

5  —  4 

+3.601819 

0.5565 

9.4435 

0  —  2 

—  .69909 

9.8446ft 

0.1554ft 

6  —  4 

+4.601819 

0.6630 

9.3370 

1  —  2 

+  .30091 

9.478423 

0.521577 

2  —  5 

-  .252274 

9.40187 

0.59813 

2  —  2 

+  1.30091 

0.11425 

9.88575 

3  —  5 

+  1.252274 

0.09770 

9.90230 

3—2 

+  2.30091 

0.36190 

9.63810 

4  —  5 

+2.252274 

0.35263 

9.64737 

4  —  2 

+3.30091 

0.5186 

9.4814 

5  —  5 

+3.252274 

0.5122 

9.4878 

5  —  2 

f4.30091 

0.6336 

9.3664 

6  —  5 

+4.252274 

0.6286 

9.3714 

0  —  3 

—1.04864 

0.02062ft 

9.97938ft 

3  —  6 

-  .902729 

9.9556 

0.0444 

1  _3!  —  .04863572 

8.6869553ft 

1.3130447ft 

4  —  6 

+  1.902729 

0.2794 

9.7206 

2  —  3 

+  .95136 

9.97835 

0.02165 

5  —  6 

+2.902729 

0.4628 

9.5372 

In  regard  to  this  table  we  may  add  that  the  form  of  the  angles  is  (ig  +  i'g')  — 
+  i>  &\  g  =  (i  +_  i'  n^\  nt.   The  differential  relative  to  the  time  is  ($  _j_  i'  7-\  ndt. 

The  preceding  table  is  applied  by  subtracting  the  logarithms  of  the  column  headed 
log.  (i  +  i'n-)9  or  by  adding  the  logarithms  of  the  column  headed  log.  (       .^). 

\ft/  *~T~*n 

We  will  now  give  the  values  of  ^,    IF,  and  -^-.,  remarking  that  in  the  inte- 

ndt  7  cosi7 

grations  the  angle  7  is  constant ;  after  the  integrations  it  changes  into  g. 


160 


A    NEW   METHOD    OF    DETERMINING 

dW 


w 


ccm 


7 

9    9' 

sin 

COS 

COS 

sin 

COS 

sin 

1 

0  —  0 

// 
-f-     3.2376 

// 
—1.2175 

// 
1.2175n£ 

n 
+  3.2376n« 

// 
—  3.0038  id 

// 

—  1.3464  nt 

1 

1  —  0 

.3901 

+  .9373 

.3901 

.9373 

-  .1287 

+     .2411 

i 

1  —  0 

+  32.6972 

-f  .0988 

-  32.6972 

+     .0988 

+  .3877 

.4802 

1 

2  —  0 

—       .2073 

-  .4647 

+       .1036 

+     .2323 

-  .0024 

+     .0114 

—  1 

2  —  0 

+       .9480 

-  .0851 

.4740 

.0425 

-  .6497 

-   1.4886 

j 

3  —  0 

+       .1350 

+  .0850 

.0450 

+     .0283 

-  .028 

+     .0801 

1 

—  2  —  1 

+       .894 

—  .167 

+       .383 

+     .07 

—  .033 

—     .10 

1 

-1  —  1 

.187 

+  .446 

.115 

.330 

—0.62 

—  1.60 

1 

0  —  1 

-  29.327 

-  .340 

-  83.900 

.973 

+  1.013 

+   1.84 

—  1 

0  —  1 

—       .015 

+  .530 

.045 

-  1.516 

—  .652 

-  1.64 

1 

1  —  1 

+     4.609 

—1.374 

7.087 

-  2.112 

+  1.264 

-  2.74 

-1 

\  I 

.670 

-  .489 

1.030 

.752 

—1.370 

—  3.21 

1 

2—  1 

.036 

+  .753 

+       .022 

+     .456 

-  .040 

+     .06 

—  1 

2  —  1 

+     7.035 

+  .061 

4.263 

+     .038 

+  .107 

—     .21 

1 

3  —  1 

.019 

-  .254 

+       .007 

+     .096 

l 

3  —  1 

+     1.431 

-  .284 

.540 

.107 

-  .296 

+     .670 

—  1 

4  —  1 

.297 

+  .105 

+       .081 

+     .029 

1 

—  1  —  2 

—  .03 

—    .11 

1 

0  —  2 

+  14.145 

+  .261 

+  20.207 

—     .373 

—1.76 

-  4.33 

1 

1  —  2 

—126.276 

4-3.459 

+419.660 

+  11.503 

-  .59 

-  1.23 

-1 

1  —  2 

.716 

-  .408 

2.380 

-  1.356 

+  .46 

+     .96 

1 

2  —  2 

1.837 

-1.119 

1.410 

.860 

+  .36 

.78 

—  1 

2  —  2 

+     9.116 

—  .475 

7.008 

.365 

-  .95 

-  2.34 

1 

3  —  2 

.470 

+  .484 

.204 

+     .210 

-  .01 

+     .01 

-1 

3^-2 

-  33.666 

-f  .990 

+   14.632 

.430 

+  .02 

—     .12 

i 

4  —  2 

.017 

-f  .125 

.005 

+     .038 

I 

4  —  2 

8.150 

-  .116 

2.469 

.035 

-  .14 

+     .30 

l 

5  —  2 

.210 

-f  .092 

+       .050 

+     .021 

1 

0  —  3 

+     1.0629 

—  .4814 

+     1.0136 

+     .4591 

—     .05 

—     .15 

1 

1  —  3 

1.5475 

-f-  .3568 

-  31.8180 

—  7.335 

-14.56 

—37.33 

l 

1  —  3 

.0120 

+  .0898 

.2452 

-  1.847 

+     .25 

—     .37 

1 

2  —  3 

—  77.4394 

+2.9904 

+  81.400 

+  3.139 

.04 

.18 

-1 

2  —  3 

+       .2972 

+  .1554 

.3124 

+     .1631 

+     .06 

+     .14 

1 

3  —  3 

3.'  ,64 

—  .7121 

+     1.679 

—     .365 

+     .13 

—     .28 

—  1 

3  —  3 

-f     2.3706 

—  .2375 

1.216 

—     .122 

.36 

.91 

1 

4  —  3 

.148 

-f  .337 

.050 

+     .115. 

.00 

.00 

—  1 

4  —  3 

—  22.377 

+  .996 

+     7.413 

+     .338-. 

.01 

—     .07 

—  1 

5  —  3 

—     2.496 

—  .047 

+       .627 

—     .012 

.06 

+     .13 

1 

1  —  4 

—       .165 

—       .414 

—     .096 

—     .29 

1 

2  —  4 

1.965  ',; 

+1.126 

+     3.265 

+  1.871 

+     .647 

+  1.71 

1 

3  —  4 

—  44.513 

+  2.479 

+  27.790 

+  1.548 

.014 

—     .05 

—  1 

3  —  4 

.031 

—  .012 

.019 

—    .007 

+     .015 

+     .03 

1 

4  —  4 

—     2.567 

—  .385 

.986 

.148 

+     .054 

—     .12 

—  1 

4  —  4 

+     1.002 

—  .963 

—       .385 

—     .370 

—     .150 

—     .40 

1 

5  —  4 

—       .022 

+  .057 

+       .006 

+     .016 

—  1 

5  —  4 

-  13.272 

+  .682 

+     3.686 

.190 

.009 

—     .04 

—  1 

6  —  4 

—    3.037 

-  .010 

+       .660 

.002 

THE    GENERAL    PERTURBATIONS    OF    THE   MINOR   PLANETS. 


161 


sin 


sin 


//                  // 

It                                                  II                                                          H                                                   " 

1 

3  —  5 

1.717       +  .195 

+           I- 

374 

+     .156 

.!<••:-;                  .4(5 

—1 

3  —  5 

+       .011         -  .017 



009 

+     .014 

.'>0!                   .01 

1 

4  —  5 

-  24.846        -j-1.626 

+       11. 

030 

+     .722 

—     .015         +     .02 

—1 

4  —  5 

.030         -  .072 

+               • 

013 

.032 

•tie            .00 

1 

5  —  5 

2.473         -  .194 

+               • 

760 

.060 

.05 

—1 

5  —  5 

+        .356         -  .080 

110 

.024 

—     .18 

1 

6  —  5 

.089       +  .160 

4      '. 

021 

+     .038 

—1 

6  —  5 

7.377       +  .556 

+     1.735 

.130 

_  1 

7  —  5 

+     1.413       +  .036 

— 

270 

+     .007 

« 

1 

4  —  6 

.964       +  .124 

.507 

+     .07 

1 

5  —  6 

-  13.223        +1.090 

-|-     4.555 

+     .38 

—  1 

5  —  6 

.167       +  .023 

+       .057 

+     .06 

1 

6  —  6 

.946         -  .002 

.242 

.00 

—1 

6  —  6 

2.098         -  .040 

.538 

.01 

—1 

7  —  6 

3.302       +  .324 

+       .674 

+      .09 

i 

The  part  of  W  independent  of  y  arising 

from  the 

factor, 

—  3,  in  the  value  of 

-4, 

has  not  yet  been  given.     Its  integral,  or  J  —  3a  \^-j 

,  is  the 

following: 

X. 

5      / 

dQ 

V 

J 

3a 

' 

V 

dg 

9    9'             cos 

sin 

9 

9' 

cos 

,  ' 

sin 

1—0         +  3.1392 

+  .8181 

4  - 

-  3 

2.74 

+  .14 

2  —  0 

+     .1509 

-  .3757 

5  - 

-3 

-     .11 

_..„; 

3-0 

.0858 

-    .1738 

2  - 

-  4 

.VI 

+  .4:5 

I  1 

.51 

+  .20 

3  - 

-  4 

-  1.54 

—.13 

j  i 

—25.39 

-  .39 

4  - 

-  4 

-1(1.74 

.tu 

2  —  1 

+  2.33 

+  .73 

5  - 

-  4 

-   1.6* 

r  .u.3 

3-    1 

.04 

—  .22 

6  - 

-  4 

's 

1  —  2 

+41.934 

+  .090 

3  - 

-5 

.14 

4.16 

2  —  2 

—91.80 

—2.53 

4 

-  5 

-     .89 

—.OB 

3  —  2 

-  4.13 

+  .34 

5  - 

-5 

-  7.49 

—.50 

4 

.10 

-  .V2 

6  - 

-5 

-     .96 

+  .02 

1  -   3 

•20 

—4.9099 

4  - 

-  6 

-     .07 

+.05 

2  —  3 

-  2,1 

5  - 

-  6 

.48 

—.04 

3  —  3 

B.4S 

6  - 

*    r, 

8.85 

—.2-7 

A.  P.  S. — VOL.  XIX.  U. 


A   NEW    METHOD    OF    DETERMINING 

Having  the  values  of  the  coefficients  of  (±  y  +  ig  +  i'g'\  both  for  IK  and  -  ~ 

^3  OOS  t/' 

we  have  next  to  find  those  of  (±vy  +  ig  +  i'g'\  and  of  (Oy  +  if)  +  i'g)  in  the  case 

c          U 

of .. 


COS  i 


The  expressions  for  this  purpose  are 


(2)     _    ±      _  _    1,$  _  1     f 

384 


__ 


For  Althaia  we  find 

log.  >?(2)  =  8.60309        log.  >7(8)  =z  7.38308        log.  >7(0)  =  9.08196/* 

We  multiply  the  coefficients  of  (dh  7  +  «#  +  *'^')  by  >7(2),  and  >y(3),  respectively, 
to  find  those  of  (±  2/  -f  ig  +  «V),     (=b  3/  +  ^^  +  *'y  ). 

In  case  of  (O/  +  /(/  +  «'^')  in  the  expression  for  -^-.  we  add  the  coefficients  of 

OOS  t 

(+  y  +  ig  +  i'g')  to  those  of  (  —  y  +  ig  +  i'g)  and  multiply  the  sum  by  »y(0). 


We  will  give  a  few  examples  to  show  the  formation  of  W,  and  —  A— - 

2  dr 

With  these  two  we  give  at  once  also  their  integrals,  which  are  n$z  and  v  respec- 
tively. 

W 


'  dY 
(0  -  0) 

u  cos  sin  sin  cos 


-  1        !-<>       —32.6972       +.0988 

-  2       2  —  0  .0190      +.0017 


—32.7162 


—  32.  7162  nt 


+  16.3486          +.0494 
+     .0190          +.0017 


+.0511 


THE   GENERAL    PERTURBATIONS    OF    THE   MINOR   PLANETS. 

!r  _id^v 


163 


(1- 

//                         // 

-  1        2  —  0         -   .474        +  .042 
0        1  —  0       +3.139        +  .818 
2_1_0        -1.314           -  .004 
1        0  —  0       —  1.2175/3*  +3.2376/1* 

-0) 

+  .2.37              +  .021 
-1.314              +  .004 

+  1.351  —  1.2l75ni     +  .856  +3.2376n£           -1.077  —  ,6087n£     +  .025  —  1.6188n£ 
+  4.59      -  1.2175n£     —2.07    —  3.2376w£           —0.54    +.6087w£     —0.58     —  1.6188n£ 

(-1 

1—2  —  1        +  .383         +  .070 
_1        0—1          -  .045         -1.516 
-  2        1  —  1          -  .041           -  .030 
0  —  1  —  1        —  .513         +  .200 

-i) 

+.191                —.035 
+  .022                —.758 
+.041                —.030 

—0.216         -1.246 
+  .16             -  .92 

+.254                —.823 
+.19                    +.61 

(i- 

-  2       3  —  1               .022        —  .004 
-  1        2  —  1             4.263         +  .038 
01  —  1       -  25.390          -  .390 
1        0  —  1       -  83.900          -  .973 

-1) 

+     .022            —.004 
+  2.131            +.019 

-41.950            +.486 

-113.574         —1.329 
—174.61          +2.04 

—39.798            +.501 
+  61.19              +0.77 

.  j  TIT 

In  the  integration  we  apply  the  proper  factor  to  each  term  of  W,  —  J  -p-,  and 

obtain  the  values  of  n&z,  v,  except  in  case  of  the  terms  (ig  +  og'). 

Let  us  take  the  term  (g  —  og')  or  (1  —  0),  and  let  ^  the  integrating  factor  to 
be  applied. 

Let  c,  a,  d,  5,  represent  the  cos,  sin,  nt  cos,  nt  sin  terms  respectively. 


164 


A    NEW    METHOD    OF    DETERMINING 


Thus  we  have 


c  d 

a  a 

+  1.351         — 1.2175n< 


a 


+.856         +3.2376n£; 


and  hence 


+1.351         +3.2376  —  1.2175w« 

or,  since  ^  is  unity, 

//  // 

+4.59  -1.2175n< 


— [id 

a 

—.856 


—2.07 


'     I'  s 

—  1.2175 


—3.2376. 


-3.2376/iJ 


In  case  of  the  term  (2  —  0),  ^  is  g. 

In  the  way  indicated  we  derive  the  values  of  nbz,  and  v.     In  the  case  of  -- 

COS  I 

we  have  the  values  at  once  without  another  integration  as  was  necessary  for  nbz  and  v. 

In  the  value  of   W  given  above  the  arbitrary  constants  of  integration  have  not 
been  applied. 

We  give  these  constants  in  the  form 


cos  y 


Then  in  case  of  —  A 

- 


sin  y 


we  have 


cos  2y  -f 


sn 


etc. 


ycL  sin  y  —  5&2  cos  y  +  >7(2)  ^  sin  2/  -  -  >?(2)  ^2  cos  2/  i  etc. 


Having  TF  from  the  integration  of  --^-,  we  form  W  from  the  value  of  W  and 
converting  y  into  g. 

We  thus  have  from  the  equation 


dz 


dt 


+  (1".351  +  Jd)  cos  g  -|-  (0".856  +  Jc2)  sin  ^ 

—  1".2175^  cos  ^  +  3".2376nt  sin  ^ 

+  (—  ".284  +  >7(2)  ^0  cos  2g  +  (0".589  +  >7(2)  &2)  sin 
-  /7.0488w?5  cos  2g  +  ".1298^^  sin  2g 

i  etc.  ±  etc. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          165 

In  the  second  integration  the  constants  of  nbz  and  v  are  designated  by  C  and  N 
respectively,  and  the  complete  forms  are 

C  +  Ictfit  +    Jci  sin  g  —  &2   cos  g  +  i>?(2)&!  sin  2g  —  J>?(2)  Jc2  cos  2g  i  etc. 
_ZV  -  pi  cos  g  -  -  p2  sin  g  -  -  J>/2)  JcL  cos  2</  —  J>?(2)  k2  sin  2#  —  etc. 

In  case  of  the  latitude  the  constants  of  integration  have  the  form 

Z0  +  h  sin  g  +  L  cos  g. 
We  thus  find 

n*  =  C  +[1  +  Aso  — 32".7162]/i< 

+  [4".59  +  A;,]  sin  g  +  [—  2".07  —  AJ  cos  g 

-  l".2175nt  sin  g  —  3".2376w^  cos  g 

+  [ — 0".ll  +  J>7(2)  k^\  sin  2^r  +  [ — 0".31  —  i>7(2)^]  cos  2g 

—  0''.0244^  sin  2cr  —  Q".QQ4Qnt  cos  2</ 

«,/  «7 

i  etc.  rb  etc. 

v  =  +  0".0511frf  +  -2V 

4  [ — 0".54  --  PJ  cos  ^  +  [ —  0".58  —  pz]  sin  ^ 

+  0".6087ri^  cos  g  —  l".6188w<  sin  # 

+  [0".05  --  i>7'2)y  cos  2g  +  [—  ".24  —  iV2)&2]  sin  2«/ 

±  etc.  =b  etc. 

u  ,  =  10  +  0".3616  +  0".3623n* 

+  [1".52  +  y  sin  ^  +  [— 0".68  +  ZJ  cos  gr 
— 1".3464^  sin  g        -  3".OQ38n*  cos  g 
+  0".32  sin  2g  -  0"16  cos  2g 

—  0".0539nt  sin  2g    —  0'M204^  cos  2g 
-4-  etc.  =b  etc. 


COS  I 


166 


A    NEW   METHOD    OF   DETERMINING 


9' 

sin 

-  0 

-0 

-f-     4.59  + 

a 

-  0 

0.11  + 

-  1 

+     3.10 

-  2 

3.00 

-  3 

+     0.23 

-  1 

-174.61 

-2 

+263.97 

-  3 

+  25.15 

-4 

+     5.71 

-  5 

+     1.64 

-  6 

+       .49 

2 

+  185.18 

-  4 

1.10 

-3 

+410.16 

-  1 

—     5.25 

-3 

-  37.24 

-  2 

+     6.77 

-4 

+       .90 

-3 

+       .92 

-5 

+       .17 

-4 

+       .34 

-  1 

+       .16 

expressions  for  i 

V                   u       '      j.    1 

m£    v              in  ta  hi 

ular  form  are  the  following  : 

•  *>**»      *    •                         J.J  1     vCv  M 

COS* 

% 

u 

V 

cos  i 

cos 

COS 

sin 

sin 

COS 

+&0  nt 
n 

+  ^ 

+Z0  +  0.36 

// 

// 

—  32.71  62w* 

+       .0511n< 

+     .3623n£ 

_  2.07  —  &2 

U.o4         -^Kl 

_     .58  _  i&2 

+  L52  +  Z, 

—     .68  +  7 

It 

" 

// 

// 

—  3.2376n£ 

+     0.6087?i£ 

-  1.6188rtf 

—  1.3464n£ 

—  3.0038n< 

// 

// 

n 

If 

g|  L-fiWfe 

+       .05  —  ^wkj 

9  J.                1  -yi(^)  l/^1 
•  24          '   77  ?y       tv 

+     .32 

—     .16 

" 

ff 

H 

// 

—     .0649w£ 

+       .0244n£ 

.0649n« 

.0539w* 

.1204w« 

-  3.09 

+     2.12 

-  1.54 

-  4.83 

-  2.03 

+  1.92 

1.30 

—     .95 

+  1.30 

+     .61 

-  1.76 

+       .12 

+     .89 

.37 

+     .25 

+  2.04 

+  61.19 

+     .77 

+  2.69 

+  1.26 

-  7.21 

—156.21 

-  4.24 

-  1.15 

—     .57 

-  0.81 

-  18.30 

.56 

-  1.60 

.60 

-  0.35 

4.68 

.29 

+     .03 

+     .02 

-  0.11 

1.45 

.09 

.05 

—       .50 

.04 

+  2.10 

-  43.27 

+     .07 

-  6.64 

—  2.70 

.71 

+       .36 

—     .01 

.47 

—    .17 

—87.44 

+  14.64 

+  3.15 

+  4.43 

+  1.73 

+     .87 

+     4.02 

+     .62 

—  1.98 

+     -«9 

+  8.03 

+  16.07 

+  3.78 

—38.24 

—14.92 

+     .04 

7.08 

.01 

.52 

+     .20 

—     .86 

1.05 

—    .70 

+  1.31 

+     .50 

+     .04 

.69 

+     .05 

.24 

+     .03 

.03 

.33' 

.04 

+     .28 

+     .10 

+     .01 

.38 

.00 

—     .92 

+       .19 

+     .61    , 

—  1.62 

—     .63 

THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS. 


167 


The  constants  of  integration  are  now  to  be  so  determined  as  to  make  the  pertur- 
bations zero  for  the  Epoch.     The  following  equations  fulfill  this  condition  : 


C 


sin  g  -      &2  cos  g  -f  J>?(2)  ^  sin  2g  —  J>y(2)  Jc2  cos 


etc. 


cos  y  +  c2  sn  g  + 
cos#  -  -  J&2  sin  ^  -  - 
sin  g  -  -  po  cos  g  + 


cos  2#  +    >?(2)  &2  sin  2#  -f  etc. 
cos  2y  -  -  ^(2)  &2  sin  2g  —  etc. 


—  0 
=  0 


sn 


cos 


etc.  +         (^)0     =  0 


+  Zi  sin  ^  +  Za  cos  g  +  >y(2)  ^  sin  2g  -{-  >?(2)  Zs  cos  2g  +  etc.  +         (-  —A    =  0 

VCOS  7-y  Q 

-  i2  sin  g 


cos  2    —^  I  sin  2r  -±-  etc. 


To  find  ^  and  k2  we  have 

[cos  g  —  e  +  V2)  cos  2#  +  >/(3)  cos  3g  +  etc.]  +  k2  [sin  ^  +  ^(^  sin  2g  +  etc.] 

-  3Z0  -f  6  (v)o  +  4-       (^)o  =  0 


[sin  ^  +  2>?(2)  sin  2g  +  3>?(3)  sin  3^  +  etc.]  —  Jc2  [cos  g  +  2^  cos  2^  +  etc.] 


where 


=  —  32".7162, 


0  being  found  from 


li. 


ndt 


6W». 


SJS 


\ 


We  have  also 


/   "V 

VQ    ~~    662. 

i 

The  symbols  (n^)0 ,  (v)0 ,  etc.,  represent  the  values  of  n$z,  v,  etc.,  at  the  Epoch. 


168  A    NEW    METHOD    OF    DETERMINING 

To  find  the  values  of  the  angles  (ig  +  i'g)  at  the  Epoch  we  have 

y  =  332°  48'  53".2 
g'  -    63     5  48  .6 

The  long  period  inequality,  5  Saturn  —  2  Jupiter,  is  included  in -the  value  of  y'. 

From  these  values  of  g  and  g'  we  find  the  various  arguments  of  the  perturbations. 
Then  forming  the  sine  and  cosine  for  each  argument,  we  multiply  the  sine  and  cosine 
coefficients  of  the  perturbations  by  their  appropriate  sines  and  cosines. 

In  forming  ~—  (nbz\  etc.,  we  can  make  use  of  the  integrating  factors,  multiply- 

itdv 

ing  by  the  numbers  in  the  column  (i  -f  *"'  -  ).    Having  their  differential  coefficients  we 
proceed  as  in  the  case  of  (?2&z),  etc. 


We  thus  find 

)0  =  +  401".7,  Wo  =  +  180".6,  (-JL)  =  —  22". G 

t  V_,S  =  —  391".6,  -4r  Wo  =  +    70".5, 

nd£  v        x  wrf^  v   "  n^ 


And  from  these  we  have 

k  =  +  412".8,        Jc.2  =  —82" .9,        7c0  -  --26".21,        10  =  0".0 

I,  =  -      45".2,         12  =  +    0".4,        JV=  +  28".3, 
(7  =        332°  44'  12".6. 

The  new  mean  motion  is  found  from  (1  —  32".7162  —  26". 21)  nt,  which  gives 
n  —  855".5196.  "With  this  value  of  n  we  find  the  only  change  is  in  the  coefficients 
of  the  argument  (1  —  3),  having  +  405".29  instead  of  410".16,  and  —  86".30  instead 
of  —  87".44. 

The  constant  G  now  has  the  value 

C  ~  332°  44'  16".3. 


THE    GENERAL    PERTURBATIONS    OF    THE    MINOR   PLANETS.  1G9 

Introducing  the  values  of  the  constants  of  integration  into  the  expressions  for 
nz9  r,  and        ,  we  have 


nz     -  332°  44'  16".3  +  855".5196  1 

+  417"  A   sing  +  80".8     cos# 

3".2376/cos# 


+    16".4      sin  20        +    3".0       cos  2^ 

0".0244  nt  sin  (2g  -      0".0649  nt  cos  2g 
db  etc.  ±  etc. 


=  +    28".3  +    0 

-  206".  9  cos  cj  +  40".9sin# 


8'  .2  cos  2g  + 


cosi 


etc.  i  etc. 


0".4  -i     0".3623n< 


1".3464  TI^  sin  ^  3".0038  nt  cos  0 

—      I".5sin20  0".2cos2<7 

0".0539  nt  sin  2r/  -     0".l  204  nt  cos  2g 

From  the  expressions  of  the  perturbations  that  have  been  given,  and  the  elements 
used  in  computing  the  perturbations,  except  that  we  use  C  in  place  of  g0  and  the  new 
value  of  the  mean  motion,  we  will  compute  a  position  of  the  body  for  the  date  1894, 
Sept.  19,  10h  48m  52s,  for  which  we  have  an  observed  position.  From  a  provisional 
ephemeris  we  have  an  approximate  value  of  the  distance;  its  logarithm  is  0.14878. 
A.  P.  s.  —  YOL.  XTX.  v. 


170  A   NEW    METHOD    OF    DETERMINING 

Reducing  the  above    date  to  Berlin  Mean  Time,  and  applying  the   aberration 
time,  we  have,  for  the  observed  date,  1894,  Sept.  19,  72800, 

g  -  339°  19'  38".l,  g'  -  65°  24'.1. 

Forming  the  arguments  of  the  perturbations  with  these,  we  find 

n&z  -  +  4'  43".2,  v  -  +  3".6,  tt  .  =  —  2".8. 


cost 


To  convert  v  into  radius  as  unity  and  in  parts  of  the  logarithm  of  the  radius 
vector  we  multiply  by  the  modulus  whose  logarithm  is  9.G3778,  and  divide  by  206264".8. 
Thus  we  have  from  v  —  +  3".6,  the  correction,  +  .000008,  to  be  applied  to  the  loga- 
rithm of  the  radius  vector. 

In  case  of          z=  —  2  '.8,  we  have 
cost 


i--  _7".19. 


Converting  into  radius  as  unity,  we  have  £z'  =  -  -  .000035.  The  coordinate  z'  is  per- 
pendicular to  the  plane  of  the  orbit.  As  we  will  use  coordinates  referred  to  the 
equator  we  have,  to  find  the  changes  in  a?,  y,  z,  due  to  a  variation  of  0',  which  we  have 
designated  by  &?',  the  following  expressions : 

fix  =  (sin  i  sin  &)  &?' 

by  —  ( —  sin  i  cos  Q,  cos  e  —  cos  i  sin  e)  $z' 

bz  =.  ( —  sin  i  cos  &  sin  e  -\-  cos  i  cos  e)  &&' 

where  e  is  the  obliquity  of  the  ecliptic. 
For  1894  we  find 

fa  =  (—  .0404)  ^',        5y  =  (—  .3123)  &',        &  =  (+.9491)  &' 

And  for  the  date  we  have 

&B  =  +  .000001        %  =  +  .000011        £*  =  —  .000033 


THE    GENERAL    PERTURBATIONS    OF    THE   MINOR   PLANETS.  171 

With          i  -  5°  44'  4".6,         Q  zz  203°  51'  51".5,        e  =  23°  27'  10".8, 
we  compute  the  auxiliary  constants  for  the  equator  from  the  formulae 

cotg  A  =  —  tg  £  cos  «,         ty  E0  =    ^  *  , 

cos  g3 

COto-  B   —        C°Si  COS  (£0  -I-  e) 

fy  Q  COS  £0    "  COS  £ 

COto-  (7  —  ____  cos*  sin  (E9  +  e) 

' 


cos 


sin  a  =  cos^,    sin  I  =   sin^cos^    sin  (7  = 
' 


sn 


sin  B  sin  (7 


The  values  of  sin  a,  sin  6,  sin  c  are  always  positive,  and  the  angle  E0  is  always 
less  than  180°. 

As  a  check  we  have 

,      .  __  sin  b  sin  c  sin  ((7  —  B) 

y  ^  —  ~" 

sin  a  cos  A 

We  find 

A  =  293°  45'  29".3,  B  =  202°  59'  46".9,  C  -  210°  45'  55".0 

log  sin  a  —  9.999645,  log  sin  I  —  9.977735,        log  sin  c  =  9.498012 

Applying  n$z  —  +  4'  43".  2  to  the  value  of  g,  we  have 

Tie  =  339°  24'  21".5 
By  means  of  g  or  nz  =  E  --  e  sin  E  we  find 

E  =  337°  39'  23".4 
Then  from 

I  sin  J  v  —  \/a(l  +  e)  sin  J  E 
*  cos  $  v  =      ttl  —  e   cos     E 


172  A    NEW    METHOD    OF    DETERMINING 

we  find 

v  -  335°  50'  12".2,        log  i\  =  0.378246 

where  v  is  the  true  anomaly. 

Calling  u  the  argument  of  the  latitude  we  have 

u  —  v  +  TI--  Q  =  143°  52'  41".8. 
Hence 

A  +  u  =  77°  38'  ll".l,        23+u  =  346°  52  28".7,         C+u  =  354°  38'  36".8. 

And  from 

x  •=.  r  sin  a  sin  (A  +  u) 
y  =  r  sin  &  sin  (13  -j-  u) 
z  •=.  r  sin  c  sin  ( C  +  w), 

where 

log  r  —  log  ?-!  +  5  log  r  =  log  r,  +  .000008, 

we  have 

x  —  +  2.331894,        y  =  —  .515433,        *==.--  .070208. 
The  equatorial  coordinates  of  the  Sun  for  the  date  of  the  observation  are 

X  -  —  1.002563         Y  -  +  .045198        Z  -  +  .019611. 
Applying  the  corrections  &c,  ^//,  ^,  we  have 
x-t-fa+X=  +  1.329332,    y  +  fy  +  F=  —  .470224,    «  +  &  +  2T  =  —  .050630. 


THE  GENERAL  PERTURBATIONS  OF  THE  MINOR  PLANETS.          173 

Then  from 

4  y  +  fy  +  y  t     \          z+  dz  +  Z     .  z  -f  3z  +  Z 

tfl  a  =r  J—  —  t  g  d  rz  -  sin  a  =  -  COS  a, 

x  -f-  8x  -f-  J£'  i/  -f  to  -f  y  a;  -f-  dx  +  ^ 


sin  8 


we  have,  giving  also  the  observed  place  for  the  purpose  of  comparison, 

* 

ttc  =  340°  31'  II"  A        &c  =  -  -  2°  3'  23".  I        log  A  rz  0.149514. 
a0  -  340   33   49.1          &0  =  -  -  2  2   25.4 

where  the  subscript  c  designates  the  computed,  and  the  subscript  o  the  observed  place. 
Both  observed  and  computed  places  are  already  referred  to  the  mean  equinox  of 
1894.0.     If  the  observed  position  were  the  apparent  place  we  should  have  to  reduce 
the  computed  also  to  apparent  place  by  means  of  the  formula) 

Aa  =  /  +  y  sin  (G  +  a)  ty  & 
AS  =          y  cos  (G  +  a), 

the  quantities/",  y,  and  G  being  taken  from  the  ephemeris  for  the  year  and  date. 

If  the  observed  position  has  not  been  corrected  for  parallax  we  refer  it  to  the  cen- 
tre of  the  Earth  by  means  of  the  formulae 

A      _          TT  p  cos  <f>'     sin  (a  —  0) 

_i  (X      •  —  '        . 

J  cos  3 

tg  a>' 
tciy  =: 

COS  (a  —  0) 
_     TT  p  *sin  <pr     sin  (y  —  S) 


A  v   _     TT  p 


sin 


where 


a  is  the  right  ascension,  $  the  declination,  A  the  distance  of  the  planet  from  the 
Earth,  $'  the  geocentric  latitude  of  the  place  of  observation,  0  the  siderial  time  of 


174          A  NEW  METHOD  OP  DETERMINING^  THE  GENERAL  PERTURBATIONS,  ETC. 

observation,  p  the  radius  of  the  Earth,  and  n  the  equatorial  horizontal  parallax  of  the 
Sun. 

For  the  difference  between  computed  and  observed  place  we  have 

C —  0  r=  —  2  37". 7  in  right  ascension,  and  C —  0  =.  -  -  57". 7  in  declination. 

By  the  method  just  given  we  have  found  the  positions  of  the  planet  for  several 
dates  and  have  compared  with  the  observed  places.  The  comparison  shows  outstand- 
ing differences  too  large  to  be  accounted  for  by  the  effects  of  the  perturbations  yet  to 
be  determined,  which  are  the  perturbations  of  the  second  order,  with  respect  to  the 
mass,  produced  by  Jupiter,  and  the  perturbations  produced  by  the  other  planets  that 
have  a  sensible  influence.  We  have  therefore  corrected  the  elements  that  have  been 
used  in  the  computations  thus  far  made,  by  means  of  differential  equations  formed  for 
this  purpose,  employing  as  the  absolute  terms  in  these  equations  the  differences  be- 
tween computation  and  observation  for  the  several  dates.  A  solution  of  the  equations 
has  given  corrections  to  the  elements  that  produce  quite  large  effects  on  the  computed 
place.  Thus  recomputing  the  position  of  the  planet  for  the  date  given  above  with  the 
corrected  elements  we  find 

ac  =  340°  33'  44".5  ,   5C  =  —  2°  2'  15".6. 

And  since 

a0  =  340°  33'  40". I  ,   80  =  —  2°  2'  25".4 
we  have,  for  the  difference  between  computed  and  observed  place, 

C —  O  =  —  4"  .6  in  right  ascension,  and  C —  O  —  +  9".8  in  declination. 


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AUG  8       1982 

Recrd  UCt  A/M/5 

JUJG  3     1981 

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